Integration by Substitution Method: Definition & Example

Integration by Substitution Method: Definition & Example

Edited By Komal Miglani | Updated on Oct 15, 2024 01:23 PM IST

Integration by substitution 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.

Integration by Substitution Method: Definition & Example
Integration by Substitution Method: Definition & Example

In this article, we will cover the concept of Integration by substitution. 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 fourteen questions have been asked on this concept, including two in 2016, one in 2018, two in 2019, three in 2020, four in 2021, one in 2022 and two in 2023.

Integration by Substitution

The method of substitution is one of the basic methods for calculating indefinite integrals. This technique transforms a complex integral into a simpler one by changing the variable of integration. It is especially useful for integrals involving composite functions where a direct integration approach is difficult.

Let f be a function of x defined on the closed interval $[\mathrm{a}, \mathrm{b}]$ and F be another function such that $\frac{d}{d x}(F(x))=f(x)$ for all $x$ in the domain of $f$, then

$$
\int_a^b f(x) d x=[F(x)+c]_a^b=F(b)-F(a)
$$

is called the definite integral of the function $f(x)$ over the interval $[a, b]$, where $a$ is called the lower limit of the integral and $b$ is called the upper limit of the integral.
Working Rule to evaluate definite Integral $\int_a^b f(x) d x$
1. First find the indefinite integration $\int f(x) d x$ and suppose the result be $\mathrm{F}(\mathrm{x})$
2. Next find the $F(b)$ and $F(a)$
3. And, finally value of definite integral is obtained by subtracting $\mathrm{F}(\mathrm{a})$ from $\mathrm{F}(\mathrm{b})$.
Thus, $\quad \int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(\mathrm{x}) \mathrm{dx}=[\mathrm{F}(\mathrm{x})]_{\mathrm{a}}^{\mathrm{b}}=\mathrm{F}(\mathrm{b})-\mathrm{F}(\mathrm{a})$.

Substitution - change of variable
To solve the integrate of the form

$I=\int f(g(x)) \cdot g^{\prime}(x) d x$

where $\mathrm{g}(\mathrm{x})$ is contimuously differentiable function.
put $\mathrm{g}(\mathrm{x})=\mathrm{t}, \mathrm{g}^{\prime}(\mathrm{x}) \mathrm{dx}=\mathrm{dt}$
After substitution, we get $\int f(t) d t$.
Evalute this integration and substitute back the value of $t$.

Some standard results using susbtitution
1. $\int \frac{f^{\prime}(\mathrm{x})}{\mathrm{f}(\mathrm{x})} \mathrm{dx}=\log _{\mathrm{e}}|\mathrm{f}(\mathrm{x})|+\mathrm{c}$
2. $\int \mathrm{f}^{\prime}(\mathrm{x})(\mathrm{f}(\mathrm{x}))^{\mathrm{n}} d \mathrm{x}=\frac{(\mathrm{f}(\mathrm{x}))^{\mathrm{n}+1}}{\mathrm{n}+1}+\mathrm{c}$

Integration of the function f(ax + b)

Integration of the function $f(a x+b)$
If $\int f(x) d x=F(x)+C$ and $a, b$ are constants, then

$\int f(a x+b) d x=\frac{1}{a} F(a x+b)+C$

we have, $I=\int f(a x+b) d x$
let $\mathrm{ax}+\mathrm{b}=\mathrm{t}$, then $\mathrm{adx}=\mathrm{d} t$

$\begin{aligned}
\therefore \quad \mathrm{I} & =\int \mathrm{f}(\mathrm{ax}+\mathrm{b}) \mathrm{dx} \\
& =\frac{1}{\mathrm{a}} \int \mathrm{f}(\mathrm{t}) \mathrm{dt} \\
& =\frac{1}{\mathrm{a}} \mathrm{F}(\mathrm{t})+\mathrm{c} \\
& =\frac{1}{\mathrm{a}} \mathrm{F}(\mathrm{ax}+\mathrm{b})+\mathrm{c}
\end{aligned}$

For example:
1. $\int \cos 2 x d x=\frac{1}{2} \sin 2 x+c$
2. $\int \frac{1}{x+1} d x=\log _e|x+1|+c$
3. $\int e^{2 x-3} d x=\frac{1}{2} e^{2 x-3}+c$

Also, Integrals of $\tan x, \cot x, \sec x, \operatorname{cosec} x$ all these can be evaluated using the result :

$\int \frac{f^{\prime}(x)}{f(x)} d x=\log |f(x)|+C$

(i)

$\begin{array}{ll}
\int \tan x d x= & \int \frac{\sec x \tan x}{\sec x} d x \\
\Rightarrow \quad & \int \tan x d x=\log |\sec x|+C
\end{array}$

(ii) $\int \cot x d x=\int \frac{\cos x}{\sin x} d x=\log |\sin x|+C$
(iii) $\int \sec x d x=\int \frac{\sec x(\sec x+\tan x)}{\sec x+\tan x} d x=\int \frac{\sec ^2 x \sec x+\tan x}{\sec x+\tan x} d x$
$\Rightarrow \quad \int \sec x d x=\log |\sec x+\tan x|+C$
(iv) $\int \csc x d x=\int \frac{\csc x(\csc x-\cot x)}{\csc x-\cot x} d x=\int \frac{\csc ^2 x-\csc x \cot x}{\csc x-\cot x} d x$

$\Rightarrow \quad \int \csc x d x=\log |\csc x-\cot x|+C$

Fundamental formulae such as $\int x^n d x=\frac{x^{n+1}}{n+1}, \int \sin x d x=-\cos x$, ,... and so on
It $x$ is replaced by a LINEAR FUNCTION of $x \Rightarrow(a x+b)$ form then,

$\int f(a x+b) d x=\frac{F(a x+b)}{\frac{\mathrm{d}}{\mathrm{d} x}(a x+b)}+c$

Recommended Video Based on Integration by Substitution


Solved Examples Based On Integration by Substitution:

Example 1: If $\int \frac{d x}{\cos ^3 x \sqrt{2 \sin 2 x}}=(\tan x)^A+C(\tan x)^B+k$, where $k$ is a constant of integration, then $A+B+C$ equals:
1) $\frac{21}{5}$
2) $\frac{16}{5}$
з) $\frac{7}{10}$
4) $\frac{27}{10}$

Solution

As learnt in the concept of Integration by substitution -

$\begin{aligned}
& \int \frac{d x}{\cos ^3 x \sqrt{2 \sin x \cos x \times 2}} \\
& =\frac{1}{2} \int \frac{\sec ^4 x d x}{\sqrt{\tan x}} \\
& =\frac{1}{2} \int \frac{\left(1+\tan ^2 x\right) \sec ^2 x d x}{\sqrt{\tan x}} \\
& =\frac{1}{2} \int(\tan x)^{\frac{-1}{2}} \sec ^2 x d x+\frac{1}{2} \int(\tan x)^{\frac{3}{2}} \sec ^2 x d x \\
& =\frac{1}{2} \frac{(\tan x)^{\frac{1}{2}}}{\frac{1}{2}}+\frac{1}{2} \frac{(\tan x)^{\frac{5}{2}}}{\frac{5}{2}}+C \\
& A=\frac{1}{2} ; B=\frac{5}{2} ; C=\frac{1}{5} \\
& A+B+C=3+\frac{1}{5} \\
& =\frac{16}{5}
\end{aligned}$


Hence, the answer is the option 2.

Example 2: The integral $\int \frac{d x}{(1+\sqrt{x}) \sqrt{x-x^2}}$ is equal to: (where C is a constant of integration.)
1) $-2 \sqrt{\frac{1+\sqrt{x}}{1-\sqrt{x}}}+C$
2) $-2 \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}+C$
3) $-\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}+C$
4) $\sqrt[2]{\frac{1+\sqrt{x}}{1-\sqrt{x}}}+C$

Solution


$\begin{aligned} & \sin x=t \Rightarrow \cos x d x=d t \\ & \int \frac{d t}{t^2\left(1+t^6\right)^{2 / 3}}=\int \frac{d t}{t^3 t^4\left(\frac{1}{t^t}+1\right)^{2 / 3}} \\ & u=\frac{1}{t^6}+1 \Rightarrow d u=-\frac{6}{t^7} d t \\ & \frac{d u}{-6}=\frac{d t}{t^7} \\ & =\int \frac{d u}{-6 u^{2 / 3}}=-\frac{1}{2} u^{1 / 3}+C \\ & =-\frac{1}{2}\left(\frac{1}{t^6}+1\right)^{1 / 3}+C \\ & =-\frac{1}{2}\left(\frac{\left(1+\sin ^6 x\right)^{1 / 3}}{\sin ^2 x}\right) \\ & f(x)=-\frac{1}{2} \frac{1}{\sin ^2 x} \quad \lambda=3 \\ & \lambda f(\pi / 3)=-2\end{aligned}$

Hence, the answer is the option 4.

Example 4: The integral $\int \frac{d x}{(x+4)^{8 / 7}(x-3)^{6 / 7}}$ is equal to: (where C ia a constant of integration)
1) $-\left(\frac{x-3}{x+4}\right)^{-1 / 7}+C$
2) $\frac{1}{2}\left(\frac{x-3}{x+4}\right)^{3 / 7}+C$
3) $\left(\frac{x-3}{x+4}\right)^{1 / 7}+C$
4) $-\frac{1}{13}\left(\frac{x-3}{x+4}\right)^{-13 / 7}+C$

Solution

$\int\left(\frac{x-3}{x+4}\right)^{\frac{-6}{7}} \frac{1}{(x+4)^2} d x$

Let $\frac{x-3}{x+4}=t^7$

$\begin{aligned}
& \frac{7}{(x+4)^2} d x=7 t^6 d t \\
& \int t^{-6} t^6 d t=t+c \\
& \left(\frac{x-3}{x+4}\right)^{\frac{1}{4}}+c
\end{aligned}$

Hence, the answer is the option 3.

Example 5: The integral $\int \frac{e^{3 \log _e 2 x}+5 e^{2 \log _e 2 x}}{e^{4 \log _e x}+5 e^{3 \log _c x}-7 e^{2 \log _e x}} d x, x>0 \quad$ is equal to : (where c is a constant of integration)
1) $\log _c\left|x^2+5 x-7\right|+c$
2) $4 \log _e\left|x^2+5 x-7\right|+c$
3) $\frac{1}{4} \log _e\left|x^2+5 x-7\right|$
4) $\log _e \sqrt{x^2+5 x-7}+c$

Solution

$\begin{aligned} & e^{\log a^x}=x \log a \\ & e^{\log _x}=x \\ & \text { Let } \mathrm{I}=\int \frac{\mathrm{e}^{3 \log _e 2 \mathrm{x}}+5 \mathrm{e}^{2 \log _e 2 \mathrm{x}}}{\mathrm{e}^{4 \log _e \mathrm{x}}+5 \mathrm{e}^{3 \log _{\mathrm{e}} \mathrm{x}}-7 \mathrm{e}^{2 \log _{\mathrm{e}} \mathrm{x}}} \mathrm{dx}, \mathrm{x}>0 \\ & \mathrm{I}=\int \frac{\mathrm{e}^{\log _e(2 x)^3}+5 \mathrm{e}^{\log _e(2 x)^2}}{\mathrm{e}^{\log _e \mathrm{x}^4}+5 \mathrm{e}^{\log _e \mathrm{x}^3}-7 \mathrm{e}^{\log _e \mathrm{x}^2}} \mathrm{dx} \\ & I=\int \frac{(2 x)^3+5(2 x)^2}{x^4+5 x^3-7 x^2} d x=\int \frac{4 x^2(2 x+5)}{x^2\left(x^2+5 x-7\right)} d x \\ & \text { put } x^2+5 x-7=t \Rightarrow(2 x+5) d x=d t \\ & I=4 \int \frac{d t}{t}=4 \ln |t|=4 \ln \left|x^2+5 x-7\right|\end{aligned}$

Hence, the answer is the option 2.

Summary:

Integration by substitution is a powerful technique used in calculus to simplify the process of finding integrals. It is an important concept in mathematics.

FAQs:

1) What is integration?

Solution: Integration is the reverse process of differentiation.

2) What is indefinite integral?

Solution: An indefinite integral, also known as an antiderivative, is a function that reverses the process of differentiation.

3) What is the other name of integration by substitution?

Solution: The other name for integration by substitution method is the Reverse Chain Rule or U-substitution method.

4) What is integration by substitution?

Solution: Integration by substitution is a method used to simplify the process of finding integrals.

5) How do you choose a suitable substitution u=g(x)u = g(x)u=g(x) for an integral?

Solution: A suitable substitution is typically chosen by identifying a part of the integrand whose derivative is also present in the integrand.

Frequently Asked Questions (FAQs)

1. What is integration?

Integration is the reverse process of differentiation.

2. What is indefinite integral?

An indefinite integral, also known as an antiderivative, is a function that reverses the process of differentiation.

3. What is the other name of integration by substitution?

The other name for integration by substitution method is the Reverse Chain Rule or U-substitution method

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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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