Integral of Particular Functions: Examples

Integral of Particular Functions: Examples

Edited By Komal Miglani | Updated on Jul 02, 2025 08:07 PM IST

Integration of indefinite integral is one of the important parts of Calculus, which applies to measuring the change in the function at a certain point. Mathematically, it forms a powerful tool by which slopes of functions are determined, the maximum and minimum of functions found, and problems on motion, growth, and decay, to name a few. These concepts of integration have been broadly applied in branches of mathematics, physics, engineering, economics, and biology.

Integral of Particular Functions: Examples
Integral of Particular Functions: Examples

In this article, we will cover the concept of Integration of indefinite integral. This concept falls under the broader category of Calculus, which is a crucial Chapter in class 12 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of two questions have been asked on this concept, including one in 2021, and one in 2023.

Integrals of Particular Function

Integration is the reverse process of differentiation. In integration, we find the function whose differential coefficient is given. The rate of change of a quantity y concerning another quantity x is called the derivative or differential coefficient of y concerning x. Geometrically, the Differentiation of a function at a point represents the slope of the tangent to the graph of the function at that point.

1. $\int \frac{d x}{x^2+a^2}=\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)+C$

$\begin{aligned}
& \text { put } x=a \tan \theta, \text { then } d x=a \sec ^2 \theta d \theta \\
& \begin{aligned}
\therefore \int \frac{d x}{x^2+a^2} & =\int \frac{a \sec ^2 \theta}{a^2+a^2 \tan ^2 \theta} d \theta \\
& =\int \frac{a^2 \sec ^2 \theta}{a^2\left(1+\tan ^2 \theta\right)} d \theta \\
& =\frac{1}{a} \int d \theta=\frac{1}{a} \theta+C=\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)+C
\end{aligned}
\end{aligned}$


2. $\int \frac{d x}{x^2-a^2}=\frac{1}{2 a} \log \left|\frac{x-a}{x+a}\right|+C$
we can rewrite above integral as

$\begin{aligned}
\int \frac{d x}{x^2-a^2} & =\frac{1}{2 a} \int\left(\frac{1}{x-a}-\frac{1}{x+a}\right) d x \\
& =\frac{1}{2 a}(\log |x-a|-\log |x+a|)+c \\
& =\frac{1}{2 a} \log \left|\frac{x-a}{x+a}\right|+C
\end{aligned}$

3. $\int \frac{d x}{a^2-x^2}=\frac{1}{2 a} \log \left|\frac{a+x}{a-x}\right|+C$

4. $\int \frac{d x}{\sqrt{a^2-x^2}}=\sin ^{-1}\left(\frac{x}{a}\right)+C$

5. $\int \frac{d x}{\sqrt{\mathrm{a}^2+\mathrm{x}^2}}=\log \left|x+\sqrt{\mathrm{x}^2+\mathrm{a}^2}\right|+C$

6. $\int \frac{d x}{\sqrt{x^2-a^2}}=\log \left|x+\sqrt{x^2-a^2}\right|+C$

7. $\int \sqrt{a^2-x^2} d x=\frac{1}{2} x \sqrt{a^2-x^2}+\frac{1}{2} a^2 \sin ^{-1}\left(\frac{x}{a}\right)+C$

8. $\int \sqrt{a^2+x^2} d x=\frac{1}{2} x \sqrt{a^2+x^2}+\frac{1}{2} a^2 \log \left|x+\sqrt{a^2+x^2}\right|+C$

9. $\int \sqrt{\mathrm{x}^2-\mathrm{a}^2} \mathrm{dx}=\frac{1}{2} \mathrm{x} \sqrt{\mathrm{x}^2-\mathrm{a}^2}-\frac{1}{2} \mathrm{a}^2 \log \left|\mathrm{x}+\sqrt{\mathrm{x}^2-\mathrm{a}^2}\right|+C$


Following are some important substitutions useful in evaluating integrals.


$\begin{array}{c|| c } \mathbf { Expression } & {\mathbf { Substitution }} \\\\ \hline \\a^{2}+x^{2} & {x=a \tan \theta} {\text { or }} {x=a \cot \theta} \\ \\\hline \\a^{2}-x^{2} & {x=a \sin \theta} {\text { or } x=a \cos \theta}\\ \\ \hline \\x^{2}-a^{2} & {x=a \sec \theta} {\text { or } x=a \csc \theta} \\\\ \hline\\ \sqrt{\frac{a-x}{a+x}} {\text { or } \sqrt{\frac{a+x}{a-x}}} & {x=a \cos 2 \theta}\\ \\\hline\end{array}$


A special type of indefinite integration:

Working rule :
for (i) put $x=a \sin \theta$ or $a \cos \theta$
for (ii) Put $x=a \sec \theta$ or $a \operatorname{cosec} \theta$
for (iii) and (iv) Put $x=a \tan \theta$ or $a \cot \theta$
for (v) and (vi) Put $x=a \cos 2 \theta$
for (vii) and (viii) Put $x=a \cos ^2 \theta+b \sin ^2 \theta$
Integrals of the form :
(i) $\int \frac{d x}{x^2+a^2}=\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)+C$
(ii) $\int \frac{d x}{x^2-a^2}=\frac{1}{2 a} \log \left|\frac{x-a}{x+a}\right|+C$
(iii) $\int \frac{d x}{a^2-x^2}=\frac{1}{2 a} \log \left|\frac{a+x}{a-x}\right|+C$
(iv) $\int \frac{d x}{\sqrt{a^2-x^2}}=\sin ^{-1}\left(\frac{x}{a}\right)+C$
(v) $\int \frac{d x}{\sqrt{a^2+x^2}}=\log \left|x+\sqrt{x^2+a^2}\right|+C$
(vi) $\int \frac{d x}{\sqrt{x^2-a^2}}=\log \left|x+\sqrt{x^2-a^2}\right|+C$

Proofs of all these formulas are as follows:

Integrals of Some Particular Functions

In this section, we mention below some important formulas of integrals and apply them for integrating many other related standard integrals:

(1) $\int \frac{d x}{x^2-a^2}=\frac{1}{2 a} \log \left|\frac{x-a}{x+a}\right|+\mathrm{C}$
proofs of all these formulas area as follows:
We have $\frac{1}{x^2-a^2}-\frac{1}{(x-a)(x+a)}$

$\equiv \frac{1}{2 a}\left[\frac{(x+a)-(x-a)}{(x-a)(x+a)}\right]-\frac{1}{2 a}\left[\frac{1}{x-a}-\frac{1}{x+a}\right]$


Therefore, $\int \frac{d x}{x^2-a^2}=\frac{1}{2 a}\left[\int \frac{d x}{x-a}-\int \frac{d x}{x+a}\right]$

$\begin{aligned}
& \left.=\frac{1}{2 a}[\log |(x-a)|-\log \mid(x+a)]\right]+\mathrm{C} \\
& =\frac{1}{2 a} \log \left|\frac{x-a}{x+a}\right|+\mathrm{C}
\end{aligned}$

(2) In view of (1) above, we have

$\frac{1}{a^2-x^2}=\frac{1}{2 a}\left[\frac{(a+x)+(a-x)}{(a+x)(a-x)}\right]=\frac{1}{2 a}\left[\frac{1}{a-x}+\frac{1}{a+x}\right]$

Therefore, $\int \frac{d x}{a^2-x^2}=\frac{1}{2 a}\left[\int \frac{d x}{a-x}+\int \frac{d x}{a+x}\right]$

$\begin{aligned}
& -\frac{1}{2 a}[-\log |a-x|+\log |a+x|]+\mathrm{C} \\
& -\frac{1}{2 a} \log \left|\frac{a+x}{a-x}\right|+\mathrm{C}
\end{aligned}$

- Note The technique used in (1) will be explained in Section 7.5.

(3) Put $x=a \tan \theta$. Then $d x=a \sec ^2 \theta d \theta$.

Therefore, $\int \frac{d x}{x^2+a^2}-\int \frac{a \sec ^2 \theta d \theta}{a^2 \tan ^2 \theta+a^2}$

(4) Let $x=a \sec \theta$. Then $d x=a \sec \theta \tan \theta d \theta$.

$-\frac{1}{a} \int d \theta=\frac{1}{a} \theta+\mathrm{C}=\frac{1}{a} \tan ^{-1} \frac{x}{a}+\mathrm{C}$


Therefore, $\quad \int \frac{d x}{\sqrt{x^2-a^2}}=\int \frac{a \sec \theta \tan \theta d \theta}{\sqrt{a^2 \sec ^2 \theta-a^2}}$
$-\int \sec \theta d \theta=\log |\sec \theta+\tan \theta|+C_1$

$=\log \left|\frac{x}{a}+\sqrt{\frac{x^2}{a^2}-1}\right|+C_1$

$-\log \left|x+\sqrt{x^2-a^2}\right|-\log |a|+C_1$

$=\log \left|x+\sqrt{x^2-a^2}\right|+C, \text { where } C=C_1-\log |a|$


(5) Let $x=a \tan \theta$. Then $d x=a \sec ^2 \theta \mathrm{d} \theta$.

$\begin{aligned}
\int \frac{d x}{\sqrt{a^2-x^2}} & =\int \frac{a \cos \theta d \theta}{\sqrt{a^2-a^2 \sin ^2 \theta}} \\
& =\int d \theta-\theta+\mathrm{C}-\sin ^{-1} \frac{x}{a}+\mathrm{C}
\end{aligned}$
Therefore, $\quad \int \frac{d x}{\sqrt{x^2+a^2}}=\int \frac{a \sec ^2 \theta d \theta}{\sqrt{a^2 \tan ^2 \theta+a^2}}$
$=\int \sec \theta d \theta-\log |(\sec \theta+\tan \theta)|+C_1$
$\begin{aligned}
& =\log \left|\frac{x}{a}+\sqrt{\frac{x^2}{a^2}+1}\right|+C_1 \\
& =\log \left|x+\sqrt{x^2+a^2}\right|-\log |a|+C_1 \\
& =\log \left|x+\sqrt{x^2+a^2}\right|+C, \text { where } C-C_1-\log |a|
\end{aligned}$

Recommended Video Based on Integrals of Particular Function


Solved Examples

Example 1: Evaluate $\int \frac{d x}{\sqrt{(x-a)(b-x)}}$

1) $2 \sin ^{-1} \sqrt{\left(\frac{x-a}{b-a}\right)}+c$

2) $2 \cos ^{-1} \sqrt{\left(\frac{x-a}{b-a}\right)}+c$

3) $2 \tan ^{-1} \sqrt{\left(\frac{x-a}{b-a}\right)}+c$

4) $2 \tan ^{-1} \sqrt{\left(\frac{x-a}{b-x}\right)}+c$

Solution

Writing $x=a \cos ^2 \theta+b \sin ^2 \theta=a+(b-a) \sin ^2 \theta$, the given integral becomes

$\begin{aligned}
& I=\int \frac{2(b-a) \sin \theta \cos \theta d \theta}{\left\{\left(a \cos ^2 \theta+b \sin ^2 \theta-a\right)\left(a \cos ^2 \theta+b \sin ^2 \theta-b\right)\right\}^{1 / 2}} \\
& =\int \frac{2(b-a) \sin \theta \cos \theta d \theta}{(b-a) \sin \theta \cos \theta}=\left(\frac{b-a}{b-a}\right) \int 2 d \theta
\end{aligned}$


$=2 \theta+c=2 \sin ^{-1} \sqrt{\left(\frac{x-a}{b-a}\right)}+c$
Hence, the answer is the option 1.

Example 2: Evaluate $\int \ln (\sqrt{1-x}+\sqrt{1+x}) d x$

1) $x \ln (\sqrt{1-x}+\sqrt{1+x})+\frac{x}{2}+\frac{1}{2} \sin ^{-1} x+c$

2) $x \ln (\sqrt{1-x}+\sqrt{1+x})-\frac{x}{2}+\frac{1}{2} \sin ^{-1} x+c$

3) $x \ln (\sqrt{1-x}+\sqrt{1+x})-\frac{x}{2}-\frac{1}{2} \sin ^{-1} x+c$

4) $x \ln (\sqrt{1-x}+\sqrt{1+x})-\frac{x}{2}+\frac{1}{2} \sin ^{-1} x+c$

Solution

We can do this question using Integration by parts

If we take

$u=\ln (\sqrt{1-x}+\sqrt{1+x}) \text { as the first function and } \mathrm{v}=1 \text { as the second function then }$


$\begin{aligned}
& \ln (\sqrt{1-x}+\sqrt{1+x}) \int 1 d x-\int\left(\frac{d}{d x}(\ln (\sqrt{1-x}+\sqrt{1+x})) \int 1 d x\right) d x \\
& =x \ln (\sqrt{1-x}+\sqrt{1+x})-\int \frac{1}{(\sqrt{1-x}+\sqrt{1+x})}\left(-\frac{1}{2 \sqrt{1-x}}+\frac{1}{2 \sqrt{1+x}}\right) x d x=x \ln (\sqrt{1-x}+\sqrt{1+x})-\frac{1}{2} \int x \frac{\sqrt{1-x^2}}{x \sqrt{1-x^2}} d x \\
& =x \ln (\sqrt{1-x}+\sqrt{1+x})-\frac{1}{2} \int d x+\frac{1}{2} \int \frac{1}{\sqrt{1-x^2}} d x
\end{aligned}$
$=x \ln (\sqrt{1-x}+\sqrt{1+x})-\frac{x}{2}+\frac{1}{2} \sin ^{-1} x+c$
Hence, the answer is the option (2).


Example 3: $\int\left(\frac{2 a+x}{a+x}\right) \sqrt{\frac{a-x}{a+x}} d x=$
1) $\sqrt{a^2-x^2}-2 a \sqrt{\frac{a-x}{a+x}}+c$
2) $-\sqrt{a^2-x^2}-2 a \sqrt{\frac{a-x}{a+x}}+c$
3) $\frac{1}{a} \tan ^{-1} \frac{x}{a}+\ln \left|x+\sqrt{a^2-x^2}\right|+c$
4) $\frac{1}{2 a} \ln \left|\frac{a+x}{a-x}\right|+\sin ^{-1} \frac{x}{a}+c$

Solution
As we learnt in
Special types of indefinite integration:
Integrals of the form:

$f\left(\sqrt{\frac{a-x}{a+x}}\right)_{\text {(ii) }} f\left(\sqrt{\frac{a+x}{a-x}}\right)$

wherein

The working rule is :

for (i) and (ii) Put $x=a \cos ($

$\begin{aligned}
& \quad \theta=\cos ^{-1} \frac{x}{a}(-a<x<a) \\
& \text { Put } I=-a \int \frac{(2+\cos \theta)(1-\cos \theta)}{1+\cos \theta} d \theta \\
& \text { and } \\
& =-a \int\left\{(1-\cos \theta)+\frac{1-\cos \theta}{1+\cos \theta}\right\} d \theta=-a\left(2 \tan \frac{\theta}{2}-\sin \theta\right)+c \\
& =\sqrt{a^2-x^2}-2 a \sqrt{\frac{a-x}{a+x}}+c
\end{aligned}$


Hence, the answer is the option 1.

Example 4: $\int \frac{d x}{x \sqrt{1-x^3}}=$
1) $\frac{1}{3} \log \left|\frac{\sqrt{1-x^3}-1}{\sqrt{1-x^3}+1}\right|+c$
2) $\frac{1}{3} \log \left|\frac{\sqrt{1-x^2}-1}{\sqrt{1-x^2}+1}\right|+c$
3) $\frac{1}{3} \log \left|\frac{1}{\sqrt{1-x^3}}\right|+c$
4) $\frac{1}{3} \log \left|1-x^3\right|+c$

Solution

As we learned,


$\int \frac{d x}{x \sqrt{1-x^3}}$


Put $1-x^3=t^2$

$\begin{aligned}
& -3 x^2 \mathrm{dx}=2 \text { tdt } \\
& =-\frac{2}{3} \int \frac{d t}{1-t^2}=\frac{1}{3} \log \left|\frac{\sqrt{1-x^3}-1}{\sqrt{1-x^3}+1}\right|+c
\end{aligned}$

Hence, the answer is the option 1.

Example 5: if $\int \sqrt{\frac{\cos x-\cos ^3 x}{\left(1-\cos ^3 x\right)}} d x=f(x)+c$, then $f(x)$ is equal to
1) $\frac{2}{3} \sin ^{-1}\left(\cos ^{3 / 2} x\right)$
2) $\frac{3}{2} \sin ^{-1}\left(\cos ^{3 / 2} x\right)$
3) $\frac{2}{3} \cos ^{-1}\left(\cos ^{3 / 2} x\right)$
4) None of these

4) None of these

Solution

As we learned in

Integration of Rational and irrational functions -

Integration in the form of :

$\begin{aligned} & \frac{d x}{\sqrt{a^2-x^2}} \\ & I=\int \sqrt{\frac{\cos x-\cos ^3 x}{1-\cos ^3 x}} d x=\int \frac{\sqrt{\cos x} \sqrt{1-\cos ^2 x}}{\sqrt{1-\left(\cos ^{3 / 2} x\right)^2}} d x \\ & \int \frac{\sqrt{\cos x} \sin x}{\sqrt{1-\left(\cos ^{3 / 2} x\right)^2}} d x \\ & \text { If } \cos ^{\frac{3}{2}} x=p, \text { then }\left(-\frac{3}{2} \cos ^{\frac{1}{2}} x \sin x\right) d x=d p \\ & I=-\frac{2}{3} \int \frac{d p}{\sqrt{1-p^2}}=-\frac{2}{3} \sin ^{-1}\left(\cos ^{\frac{3}{2} x}\right)=\frac{2}{3} \cos ^{-1}\left(\cos ^{\frac{3}{2}} x\right)+c_1\end{aligned}$

Hence, the answer is the option 3.

Summary

An indefinite integral, also known as an antiderivative, is a function that reverses the process of differentiation. It's an important concept of the calculus. In physics, integration is used to calculate quantities such as work, energy, and centre of mass.

Frequently Asked Questions (FAQs)

1. What's the connection between the integrals of sine and cosine functions?
The integral of sin(x) is -cos(x) + C, while the integral of cos(x) is sin(x) + C. These integrals are closely related, differing only by a negative sign. This relationship reflects the cyclic nature of trigonometric functions and their derivatives.
2. Why do we need absolute value signs in the integral of 1/x?
The integral of 1/x is ln|x| + C, with absolute value signs around x. This is because the natural logarithm is only defined for positive numbers, but 1/x is defined for all non-zero real numbers. The absolute value ensures that the result is defined for both positive and negative x values.
3. How does the integral of tan(x) relate to ln|sec(x)|?
The integral of tan(x) is -ln|cos(x)| + C, which can also be written as ln|sec(x)| + C. This relationship isn't immediately obvious but can be derived using substitution. It demonstrates the interconnectedness of trigonometric and logarithmic functions in calculus.
4. What's special about the integral of sec^2(x)?
The integral of sec^2(x) is tan(x) + C. This is notable because sec^2(x) is the derivative of tan(x), making this integral-derivative pair particularly easy to remember. It's often used in trigonometric substitutions and related rates problems.
5. Why doesn't the power rule for integration work for ∫(1/x)dx?
The power rule for integration states that ∫x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1. For 1/x, n = -1, which would lead to a division by zero in the formula. This special case requires a different approach, resulting in the natural logarithm function: ∫(1/x)dx = ln|x| + C.
6. What is the significance of integrating particular functions in calculus?
Integrating particular functions is crucial in calculus as it allows us to find antiderivatives of common functions, which are essential for solving real-world problems involving areas, volumes, and accumulation. These integrals form the building blocks for more complex integration techniques and applications.
7. How does the integral of a constant function differ from other functions?
The integral of a constant function c is cx + C, where C is the constant of integration. This is unique because it's the only case where integrating adds a variable (x) that wasn't present in the original function. For all other functions, integration typically increases the power of x or introduces a transcendental function.
8. What is the relationship between the integral of x^n and the power rule for derivatives?
The integral of x^n (where n ≠ -1) is (x^(n+1))/(n+1) + C. This is directly related to the power rule for derivatives, which states that the derivative of x^n is nx^(n-1). The integral essentially "undoes" the derivative by increasing the power by 1 and dividing by the new power.
9. Why is the integral of 1/x treated differently from other power functions?
The integral of 1/x (or x^-1) is ln|x| + C, which doesn't follow the pattern of other power functions. This is because if we applied the standard power rule for integration, we'd get x^0/0, which is undefined. The natural logarithm function emerges as the unique solution to this integral.
10. How does the integral of e^x relate to its derivative?
The integral of e^x is e^x + C. This is unique because e^x is the only function (up to a constant multiple) that is its own derivative and integral. This property makes e^x particularly important in calculus and differential equations.
11. Why do we need to memorize the integrals of basic functions?
We don't necessarily need to memorize integrals of basic functions. Instead, understanding the concepts and patterns behind these integrals is more important. Recognizing the relationship between derivatives and integrals can help you derive many integrals on your own. However, familiarity with common integrals can save time in problem-solving.
12. How does the integral of a sum of functions relate to the sum of their individual integrals?
The integral of a sum of functions is equal to the sum of their individual integrals. This property, known as the linearity of integration, allows us to break down complex integrals into simpler parts. For example, ∫(f(x) + g(x))dx = ∫f(x)dx + ∫g(x)dx.
13. What's the significance of the constant of integration (C) in indefinite integrals?
The constant of integration (C) represents the family of all antiderivatives for a given function. It accounts for the fact that the derivative of a constant is zero, so any constant can be added to an antiderivative without changing its derivative. This is why indefinite integrals always include "+ C".
14. How does the integral of a composite function relate to the chain rule?
The integral of a composite function often requires the reverse application of the chain rule, which is achieved through u-substitution. For example, to integrate ∫f(g(x))g'(x)dx, we might substitute u = g(x), leading to ∫f(u)du. This process essentially "undoes" the chain rule used in differentiation.
15. How does the method of partial fractions relate to integrating rational functions?
Partial fractions decomposition is a technique used to integrate complex rational functions by breaking them down into simpler fractions. Each of these simpler fractions typically has an known integral form. This method is especially useful for integrating functions like (x^2 + 1)/(x^3 - x).
16. How does the integral of sec^3(x) relate to that of sec(x)tan(x)?
The integral of sec^3(x) is (1/2)(sec(x)tan(x) + ln|sec(x) + tan(x)|) + C. This is more complex than the integral of sec(x)tan(x) and requires integration by parts or a clever substitution. It shows how increasing the power of a trigonometric function can significantly complicate its integral.
17. How does the integral of x^n * e^x relate to integration by parts?
The integral of x^n * e^x is typically solved using integration by parts, often repeatedly. This process generates a recursive formula: ∫x^n * e^x dx = x^n * e^x - n∫x^(n-1) * e^x dx. It's a classic example of how integration by parts can be used systematically to solve complex integrals.
18. How are the integrals of sin^2(x) and cos^2(x) related?
The integrals of sin^2(x) and cos^2(x) are related through the identity sin^2(x) + cos^2(x) = 1. Their integrals are:
19. Why is the integral of sec(x) more complex than other basic trigonometric functions?
The integral of sec(x) is ln|sec(x) + tan(x)| + C, which is more complex than other basic trigonometric integrals. This complexity arises because sec(x) doesn't have a simple relationship to simpler functions like sin(x) or cos(x) in terms of derivatives. Its integration often requires a clever substitution or a complex trigonometric identity.
20. How does the integral of 1/(1+x^2) relate to the arctangent function?
The integral of 1/(1+x^2) is arctan(x) + C. This relationship defines the arctangent function and is crucial in many applications, including those involving Cauchy distributions in probability theory. It's a prime example of how integration can lead to the introduction of new functions in mathematics.
21. What's the connection between the integrals of csc(x) and sec(x)?
The integrals of csc(x) and sec(x) are related but distinct:
22. How does the integral of e^(ax) relate to the integral of e^x?
The integral of e^(ax) is (1/a)e^(ax) + C, where a is a constant. This is related to the integral of e^x (which is e^x + C) through a simple substitution and the chain rule. The factor 1/a appears due to the chain rule, demonstrating how constants in the exponent affect the antiderivative.
23. Why is the integral of 1/sqrt(1-x^2) equal to arcsin(x)?
The integral of 1/sqrt(1-x^2) is arcsin(x) + C. This relationship defines the arcsine function and is derived from the derivative of sin(x) using a clever substitution. It's crucial in solving problems involving circular motion and trigonometric substitutions.
24. What's the significance of the integral of 1/(x^2 + a^2) in relation to inverse trigonometric functions?
The integral of 1/(x^2 + a^2) is (1/a)arctan(x/a) + C. This integral is significant as it introduces the arctangent function and is closely related to the integral of 1/(1+x^2). It's often used in problems involving Cauchy distributions and in solving certain types of differential equations.
25. How does the integral of ln(x) differ from other elementary functions?
The integral of ln(x) is x ln(x) - x + C. This integral is unique because it combines algebraic and logarithmic functions. It doesn't follow the pattern of other elementary functions and is often solved using integration by parts, demonstrating the power of this technique.
26. Why is the integral of sec(x)tan(x) simpler than the integrals of sec(x) or tan(x) individually?
The integral of sec(x)tan(x) is sec(x) + C. This is simpler than the integrals of sec(x) or tan(x) individually because sec(x)tan(x) is the derivative of sec(x). This makes it a straightforward application of the fundamental theorem of calculus, unlike the more complex integrals of sec(x) and tan(x) alone.
27. How does the integral of cosh(x) relate to that of sinh(x)?
The integral of cosh(x) is sinh(x) + C, while the integral of sinh(x) is cosh(x) + C. This relationship mirrors that of sine and cosine, reflecting the deep connections between hyperbolic and trigonometric functions. The similarity in their integrals underscores the analogous properties of these function pairs.
28. What's the significance of the integral of 1/x^2 in physics and mathematics?
The integral of 1/x^2 is -1/x + C. This integral is significant in physics for describing gravitational and electromagnetic fields, and in mathematics for its role in convergence tests for infinite series. It's also an example where integration decreases the power of x, contrary to most power functions.
29. How does the integral of a^x relate to the integral of e^x?
The integral of a^x is (a^x)/ln(a) + C, where a > 0 and a ≠ 1. This is related to the integral of e^x through the change of base formula: a^x = e^(x ln(a)). The factor 1/ln(a) appears due to the chain rule, showing how the base of the exponential affects the antiderivative.
30. Why is the integral of sin(x)cos(x) equal to -cos(2x)/2?
The integral of sin(x)cos(x) is -cos(2x)/2 + C. This result comes from the trigonometric identity sin(2x) = 2sin(x)cos(x). By recognizing this pattern, we can simplify the integral and introduce a function with double the original angle, demonstrating the interconnectedness of trigonometric functions and their integrals.
31. How does the integral of 1/(ax+b) relate to natural logarithms?
The integral of 1/(ax+b) is (1/a)ln|ax+b| + C. This is a generalization of the integral of 1/x and shows how linear functions in the denominator lead to natural logarithms in the antiderivative. The constant 1/a appears due to the chain rule, reflecting the impact of the coefficient of x on the result.
32. What's special about the integral of e^(x^2)?
The integral of e^(x^2) cannot be expressed in terms of elementary functions. This integral, known as the error function (when properly scaled), is an example of a function that requires special functions or numerical methods to evaluate. It demonstrates that not all seemingly simple integrals have elementary antiderivatives.
33. How does the integral of sqrt(x) relate to the power rule for integration?
The integral of sqrt(x) is (2/3)x^(3/2) + C. This follows the general power rule for integration, where we add 1 to the exponent (1/2 becomes 3/2) and divide by the new exponent. It demonstrates how the power rule applies to fractional exponents, not just integers.
34. Why is the integral of tan^2(x) not as straightforward as other trigonometric functions?
The integral of tan^2(x) is tan(x) - x + C. This isn't as straightforward as other trigonometric integrals because it requires the use of the identity tan^2(x) = sec^2(x) - 1, followed by integrating sec^2(x) and x separately. It showcases the importance of trigonometric identities in integration.
35. How does the integral of 1/(x ln(x)) relate to the natural logarithm function?
The integral of 1/(x ln(x)) is ln|ln|x|| + C. This integral, known as the logarithmic integral, introduces a "double logarithm" and is important in number theory, particularly in the study of the distribution of prime numbers. It demonstrates how nested logarithms can arise from integration.
36. What's the significance of the integral of sin(x)/x in signal processing?
The integral of sin(x)/x, while not expressible in terms of elementary functions, is crucial in signal processing. Its definite integral from -∞ to ∞ equals π, and the function sinc(x) = sin(x)/x plays a key role in sampling theory and Fourier transforms. This integral demonstrates the importance of non-elementary functions in applied mathematics.
37. Why is the integral of 1/(1-x^2) related to both arctanh(x) and arcsin(x)?
The integral of 1/(1-x^2) is arctanh(x) + C for |x| < 1, and arcsin(x) + C for |x| ≤ 1. This dual nature arises from the relationship between hyperbolic and circular trigonometric functions. It demonstrates how the domain of the function can affect the form of its antiderivative.
38. How does the integral of csc^2(x) relate to the derivative of cot(x)?
The integral of csc^2(x) is -cot(x) + C. This relationship is significant because csc^2(x) is the negative derivative of cot(x). It's a prime example of how recognizing derivative relationships can simplify integration, similar to the integral of sec^2(x) being tan(x) + C.
39. What's the connection between the integrals of sin(x^2) and cos(x^2)?
The integrals of sin(x^2) and cos(x^2) cannot be expressed in terms of elementary functions. They are related to Fresnel integrals, which are important in optics and signal processing. These integrals demonstrate that even simple-looking functions can have complex, non-elementary antiderivatives.

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Arrange the following Cobalt complexes in the order of incresing Crystal Field Stabilization Energy (CFSE) value. Complexes :  

\mathrm{\underset{\textbf{A}}{\left [ CoF_{6} \right ]^{3-}},\underset{\textbf{B}}{\left [ Co\left ( H_{2}O \right )_{6} \right ]^{2+}},\underset{\textbf{C}}{\left [ Co\left ( NH_{3} \right )_{6} \right ]^{3+}}\: and\: \ \underset{\textbf{D}}{\left [ Co\left ( en \right )_{3} \right ]^{3+}}}

Choose the correct option :
Option: 1 \mathrm{B< C< D< A}
Option: 2 \mathrm{B< A< C< D}
Option: 3 \mathrm{A< B< C< D}
Option: 4 \mathrm{C< D< B< A}

The type of hybridisation and magnetic property of the complex \left[\mathrm{MnCl}_{6}\right]^{3-}, respectively, are :
Option: 1 \mathrm{sp ^{3} d ^{2} \text { and diamagnetic }}
Option: 2 \mathrm{sp ^{3} d ^{2} \text { and diamagnetic }}
Option: 3 \mathrm{sp ^{3} d ^{2} \text { and diamagnetic }}
Option: 4 \mathrm{sp ^{3} d ^{2} \text { and diamagnetic }}
Option: 5 \mathrm{d ^{2} sp ^{3} \text { and diamagnetic }}
Option: 6 \mathrm{d ^{2} sp ^{3} \text { and diamagnetic }}
Option: 7 \mathrm{d ^{2} sp ^{3} \text { and diamagnetic }}
Option: 8 \mathrm{d ^{2} sp ^{3} \text { and diamagnetic }}
Option: 9 \mathrm{d ^{2} sp ^{3} \text { and paramagnetic }}
Option: 10 \mathrm{d ^{2} sp ^{3} \text { and paramagnetic }}
Option: 11 \mathrm{d ^{2} sp ^{3} \text { and paramagnetic }}
Option: 12 \mathrm{d ^{2} sp ^{3} \text { and paramagnetic }}
Option: 13 \mathrm{sp ^{3} d ^{2} \text { and paramagnetic }}
Option: 14 \mathrm{sp ^{3} d ^{2} \text { and paramagnetic }}
Option: 15 \mathrm{sp ^{3} d ^{2} \text { and paramagnetic }}
Option: 16 \mathrm{sp ^{3} d ^{2} \text { and paramagnetic }}
The number of geometrical isomers found in the metal complexes \mathrm{\left[ PtCl _{2}\left( NH _{3}\right)_{2}\right],\left[ Ni ( CO )_{4}\right], \left[ Ru \left( H _{2} O \right)_{3} Cl _{3}\right] \text { and }\left[ CoCl _{2}\left( NH _{3}\right)_{4}\right]^{+}} respectively, are :
Option: 1 1,1,1,1
Option: 2 1,1,1,1
Option: 3 1,1,1,1
Option: 4 1,1,1,1
Option: 5 2,1,2,2
Option: 6 2,1,2,2
Option: 7 2,1,2,2
Option: 8 2,1,2,2
Option: 9 2,0,2,2
Option: 10 2,0,2,2
Option: 11 2,0,2,2
Option: 12 2,0,2,2
Option: 13 2,1,2,1
Option: 14 2,1,2,1
Option: 15 2,1,2,1
Option: 16 2,1,2,1
Spin only magnetic moment of an octahedral complex of \mathrm{Fe}^{2+} in the presence of a strong field ligand in BM is :
Option: 1 4.89
Option: 2 4.89
Option: 3 4.89
Option: 4 4.89
Option: 5 2.82
Option: 6 2.82
Option: 7 2.82
Option: 8 2.82
Option: 9 0
Option: 10 0
Option: 11 0
Option: 12 0
Option: 13 3.46
Option: 14 3.46
Option: 15 3.46
Option: 16 3.46

3 moles of metal complex with formula \mathrm{Co}(\mathrm{en})_{2} \mathrm{Cl}_{3} gives 3 moles of silver chloride on treatment with excess of silver nitrate. The secondary valency of CO in the complex is_______.
(Round off to the nearest integer)
 

The overall stability constant of the complex ion \mathrm{\left [ Cu\left ( NH_{3} \right )_{4} \right ]^{2+}} is 2.1\times 10^{1 3}. The overall dissociation constant is y\times 10^{-14}. Then y is ___________(Nearest integer)
 

Identify the correct order of solubility in aqueous medium:

Option: 1

Na2S > ZnS > CuS


Option: 2

CuS > ZnS > Na2S


Option: 3

ZnS > Na2S > CuS


Option: 4

Na2S > CuS > ZnS


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