Integration
The antiderivative of $g’(x)$ with respect to $dx$ is found through the process of integration, and is given by:
∫ g’(x) \ dx = g(x) + C, \ where \ C \ is \ the \ constant \ of \ integration.
The two types of integrals include:
- Indefinite Integrals
- Definite Integrals
Definite Integral: An integral with specified upper and lower limits, without the constant of integration.
Indefinite integral: An integral without limits and an arbitrary constant added.
This article provides a comprehensive overview of standard integrals, their properties, important formulas, and examples of integration, which will help students gain a deeper understanding of the topic.
Standard Integrals
Integrals of Rational and Irrational Functions
(\begin{array}{l} \int x^n , dx = \frac{x^{n+1}}{n+1} + C, \quad n \ne 1 \ \int \frac{1}{x} , dx = \ln |x| + C \ \int c , dx = c \cdot x + C \ \int x , dx = \frac{x^2}{2} + C \ \int x^2 , dx = \frac{x^3}{3} + C \ \int \frac{1}{x^2} , dx = -\frac{1}{x} + C \\end{array})
(∫√x,dx=2⋅x⋅√x3+C ∫11+x2,dx=arctanx+C ∫1√1−x2,dx=arcsinx+C)
Integrals of Trigonometric Functions
(\begin{array}{l} \int \sin x,dx = -\cos x + C \ \int \cos x,dx = \sin x + C \ \int \tan x,dx = \ln|\sec x| + C \ \int \sec x,dx = \ln|\tan x + \sec x | + C \\end{array})
(∫sin2x,dx=12(x−sinx⋅cosx)+C ∫cos2x,dx=12(x+sinx⋅cosx)+C ∫tan2x,dx=tanx−x+C ∫sec2x,dx=tanx+C)
Integrals of Exponential and Logarithmic Functions
(∫lnx,dx=x⋅lnx−x+C ∫xn⋅lnx,dx=xn+1⋅lnxn+1−xn+1(n+1)2+C ∫ex,dx=ex+C ∫ax,dx=axlna+C )
Properties of Integration
Property 1: (\int\limits_{a}^{a} f(x),dx = 0)
Property 2: (\int\limits_{a}^{b}{f(x),dx=-\int\limits_{b}^{a}{f(x),dx}})
Property 3: (\int\limits_{a}^{b}{f(x) , dx} = \int\limits_{a}^{b}{f(t) , dt})
Property 4: (\int\limits_{a}^{b}{f(x),dx} = \int\limits_{a}^{c}{f(x),dx} + \int\limits_{c}^{b}{f(x),dx})
Property 5: (\int\limits_{a}^{b}{f(x)dx} = \int\limits_{a}^{b}{f(a+b-x)dx})
(a∫0f(x)dx=a∫0f(a−x)dx)
⇒ Also Read Definite and Indefinite Integration
Useful Formulas
* (\int{{e}^{ax}}\sin bx=\frac{{e}^{ax}}{{a}^{2}+{b}^{2}}\left[ a\sin bx-b\cos bx \right])
* (\int{{{e}^{ax}}\cos bx=\frac{{{e}^{ax}}}{{{a}^{2}}+{{b}^{2}}}\left[ a\cos bx+b\sin bx \right]}\)
*(\int{{e}^{x}}\left( f(x)+f’(x) \right) = {e}^{x}f(x))
Illustration:
(\int{{e}^{x}}(\sin x+\cos x)dx={{e}^{x}}\sin x+c)
(\int{{e}^{x}(lnx+\frac{1}{x})dx={{e}^{x}}lnx+c)
Integrating Trigonometric Functions
Type 1: (\int{{{\sin }^{m}}x{{\cos }^{n}}xdx})
- If m is odd, put cos x = t
2. If n is odd, let sin x = t
If m and n are rational, then put tan x = t
If both are even, then use the reduction method
(Q∫(1−t2)t6dt=∫cos3xsin6xdx )
t = sin(x)
(\int{{{t}^{-5}}dt})
(=−15sin5x+13sin3x+c)
Type 2: (\int{\frac{dx}{a\cos x + b\sin x + c}})
t = \tan\left(\frac{x}{2}\right)
Illustration
(∫dx2+sinx=∫d(2tan(x2))2+sinx )
(⇒∫d(2tan(x2))2+sinx=∫2dt1+t2 )
(⇒t=tan(x2) )
(\frac{d}{dt}\left( {{t}^{2}} \right)=2t)
(=∫dtt2+t+1)
(\frac{2}{\sqrt{3}}\arctan\left(\frac{2t+1}{\sqrt{3}}\right))
(=2√3arctan(2tanx2+1√3)+c)
Substitutions for Irrational Functions
Form 1: (\displaystyle \int \sqrt{Quadratic} \ dx )
(Substitute m=Qudratic, n=Linear)
Form 2: (\int{\frac{dx}{\sqrt{l_{1}in}}},,\int{\frac{l_{1}in}{\sqrt{l_{1}in}}}dx,,\int{\frac{\sqrt{l_{1}in}}{l_{1}in}dx})
(Substitute lin1=t2 → lin1=t2 )
**Form 3:** (\int{\frac{1}{\sqrt{Qua}}dx})
Substitute lin=1t
Form 4: (\int{\frac{dx}{\left( a{{x}^{2}}+b \right)\sqrt{\left( {{x}^{2}}+d \right)}})
Substitute x = $\frac{1}{t}$ and then $u^2$ for $a t^2 + b$
Integration Formulas
-
(\int\limits_{a}^{b}{f(x),dx} = \int\limits_{a}^{b}{f(t),dt})
-
(−b∫af(x)dx=a∫bf(x)dx )
∫baf(x),dx=∫caf(x),dx+∫bcf(x),dx
-
(\int\limits_{a}^{b}{f(x)dx=\int\limits_{a}^{b}{f(a+b-x)dx}})
-
2a∫0f(x)dx=a∫0f(x)dx+a∫0f(2a−x)dx =0iff(2a−x)=−f(x) and =2a∫0f(x)iff(2a−x)=f(x)
6. ∫a−af(x)dx={0if f(x) is odd 2∫a0f(x)dxif f(x) is even
Problems on Integration
Illustration:
∫20x2[x]dx=∫10x2[x]dx+∫21x2[x]dx
(\int\limits_{0}^{1}{x^2,dx} + \int\limits_{1}^{2}{x^2,dx})
$\int_{1}^{2}\frac{x^3}{3}dx = 0$
(\frac{8-1}{3} = \frac{7}{3})
Illustration:
(\int\limits_{{\pi }/{6}}^{{\pi }/{3}}{\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx})
(I=∫√cos(π2–x)√sin(π2–x)+√cos(π2–x)dx)
(\int\limits_{\frac{\pi}{6}}^{\frac{\pi}{3}}{\frac{\sqrt{\sin x} + \sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}},dx} = \int\limits_{\frac{\pi}{6}}^{\frac{\pi}{3}}{1,dx} = \frac{\pi}{6})
(I=π12)
Illustration:
(I=∫sin100xcos99x,dx )
Here f(2π - x) = f(x)
(Or\ I=2π∫0sin100xcos99x)
(\int\limits_{0}^{\pi }{{{\sin }^{100}}\left( \pi -x \right){{\cos }^{99}}\left( \pi -x \right)},dx = 2)
-I = I
I = 0
Illustration:
(\int\limits_{-5}^{5}{{{x}^{3}}\ \text{d}x=0} \ \text{as}\ f(x)=x^3 \ \text{is\ an\ odd\ function})
Leibnitz’s Rule
(\frac{d}{dx}\int\limits_{u(x)}^{v(x)}{f(t)dt} = f(v(x))\frac{dv(x)}{dx} - f(u(x))u’(x))
Practice Problems
Problem 1. (If x3∫x21logtdt=y, find dydx)
(\frac{dy}{dx}=x\left( x-1 \right){{\left( \log x \right)}^{-1}})
Problem 2. If \int\limits_{\sin x}^{1}{{{t}^{2}}f\left( t \right)dt=1-\sin x. where $x \in (0, \frac{\pi}{2})$, find $f(\frac{1}{\sqrt{3}})$.
Problem 3. (\displaystyle \lim_{x \to 2} \int_{6}^{f(x)}\frac{4t^{3}}{x-2}dt = 18.)
Problem 4. (\displaystyle \lim_{x\to \infty }\frac{\int\limits_{0}^{x}{{{e}^{{{x}^{2}}}}dx}}{\int\limits_{0}^{x}{{{e}^{2{{x}^{2}}}}dx}}=0)
Integration by Parts
(\int{uv,dx} = u\int{vdx} - \int{u’\left( \int{vdx} \right)},dx)
Illustration:
Q. (\int{\ln,x,dx} = \int{\ln,x.1,dx})
(x,ℓn,x−∫x,dx−∫1x,dx)
(=ℓn,x−1)
Q. (\int x,{{e}^{x}}dx = x\int{{{e}^{x}}dx - \int{{{\left( 1 \right)}}\left( \int{{{e}^{x}}dx} \right)}dx} )
(\int x{{e}^{x}}dx = x{{e}^{x}} - \int{{{e}^{x}}dx})
\(\frac{d}{dx} \left( xe^x - e^x \right) \)
Integration of Irrational Algebraic Functions
Type $\int{\frac{dx}{{{\left( ax+b \right)}^{k}}\sqrt{px+q}}};$
Q. (\int{\frac{x}{\left( x-3 \right)\sqrt{x+1}}dx})
x + 1 = t2
$\Rightarrow$ x = t2 - 1
(I=2∫t2−1t2−4,dt)
\(\int{2}+\frac{3}{{{t}^{2}}-4}dt\)
(2t+32ln|t−2t+2|+c)
(2√x+1+32ln|√x+1−2√x+1+2|+c)
(\int\limits_{0}^{2a}{f\left( x \right)dx} = \int\limits_{0}^{a}{f\left( x \right)dx} + \int\limits_{0}^{a}{f\left( 2a-x \right)dx})
f(2a - x) = -f(x) \Rightarrow 0
(=2a∫0f(2a−x) if f(2a–x)=f(x) )
Optimizing Area for Maximum and Minimum Values
Illustration:
f(x) = x^2 + 2
I love to listen to music!
Answer: I enjoy listening to music!
(\int\limits_{2}^{\alpha }{\left( {{x}^{2}}+2-f\left( x \right) \right)dx} ={{\alpha }^{3}}-4{{\alpha }^{2}}+8 )
Differentiating the Labniz Equation
(3α2−8α−α2−2+f(α)=0)
$f(x) = -2x^2 + 8x + 2$
Important JEE Main Questions for Integrations
Definite Integration JEE Questions
![Definite Integration]()
Indefinite Integration JEE Questions
![Indefinite Integration]()
Solved Problems on Integration
Problem 1: (\int_{a}^{b}{\frac{dx}{\cos (x-a)\cos (x-b)}})
Given:
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Solution:
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(\begin{array}{l}
\int_{}^{{}}{\frac{dx}{\cos(x-a)\cos(x-b)}}\
= \frac{1}{\sin(a-b)}\int_{}^{{}}{\frac{\sin\left{(x-b)-(x-a)\right}}{\cos(x-a),.,\cos(x-b)},dx}\
= \frac{1}{\sin(a-b)}\int_{}^{{}}{\left{\frac{\sin(x-b)}{\cos(x-b)}-\frac{\sin(x-a)}{\cos(x-a)}\right}dx}\
= \text{cosec},(a-b)\log\frac{\cos(x-a)}{\cos(x-b)}+c
\end{array})
Problem 2: (\int_{}^{}\frac{dx}{\sqrt{x+a}+\sqrt{x+b}})
Given:
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Solution:
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(\begin{array}{l} \int_{{}}^{{}}{\frac{dx}{\sqrt{x+a}+\sqrt{x+b}} = \int_{{}}^{{}}{\frac{\sqrt{x+a}-\sqrt{x+b}}{(x+a)-(x+b)},dx} \\ = \frac{1}{(a-b)}\int_{{}}^{{}}{{{(x+a)}^{1/2}},dx} - \frac{1}{(a-b)}\int_{{}}^{{}}{{{(x+b)}^{1/2}},dx} \\ = \frac{2}{3(a-b)}[{{(x+a)}^{3/2}}-{{(x+b)}^{3/2}}] + c \end{array})
Problem 3: (\int_{}^{}{\frac{x^3-x-2}{(1-x^2)} \ dx})
Given:
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Solution:
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∫x3−x−2(1−x2),dx=∫−x(1−x2)(1−x2),dx−∫21−x2,dx =−∫x,dx−2∫11−x2,dx =−x22+log(x−1x+1)+c.
Problem 4: (\int_{}^{}\frac{{{\sin }^{8}}x-{{\cos }^{8}}x}{1-2{{\sin }^{2}}x{{\cos }^{2}}x}\ dx)
Given:
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Solution:
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(∫sin8x−cos8x1−2sin2xcos2x,dx =∫(sin4x+cos4x)(sin4x−cos4x)(sin2x+cos2x)2−2sin2xcos2x,dx =∫(sin4x−cos4x),dx =∫(sin2x+cos2x)(sin2x−cos2x),dx =∫(sin2x−cos2x),dx =∫−cos2x,dx=−sin2x2+c)
Problem 5: (\int_{}^{}{\frac{x^2 , dx}{(a+bx)^2}} = )
Given:
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Solution:
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x = (t - a) / b
dx = dt/b
(I=1b2[x+ab−2ablog(a+bx)−a2b1(a+bx)])
Problem 6: Solve (\int{\frac{2\cos x+3\sin x}{4\cos x+5\sin x}},dx)
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Solution:
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Problem of type ∫acosx+bsinx+pccosx+dsinx+q,dx,∫aex+be−x+cdex+fe−x+h,dx can be solved by Nr=nDr+mDr′ she said
She said, “Now,”
2 cos x + 3 sin x = a(4 cos x + 5 sin x) + b(-4 sin x + 5 cos x)
Solving by comparing, we get
(a=2341 b=−241 )
(\therefore I=\int{\frac{25}{41}\left( \frac{-4\sin x+5\cos x}{4\cos x+5\sin x} \right)dx})
(2341x−241ln|4cosx+5sinx|+c)
Problem 7: Find the area of the region bounded by the curves $y^2 \leq 4x$, $x^2 + y^2 \geq 2x$, and $x \leq y + 2$ in the first quadrant.
Given:
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Answer:
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(\int\limits_{0}^{{{\left( \sqrt{3}+1 \right)}^{2}}}{\sqrt{4x},,dx-} ar(semicircle) + ar(\triangle ABC))
(\frac{2{{x}^{3/2}}}{3}_{0}^{{{\left( \sqrt{3}+1 \right)}^{2}}}-\frac{1}{2}{{\left( \sqrt{3}+1 \right)}^{2}}2\left( \sqrt{3}+1 \right)-\sqrt{4}\left( \frac{\pi }{2} \right))
(\frac{{{\left( \sqrt{3}+1 \right)}^{3}}{3} - \frac{\pi}{2})
Most Important Questions from Definite Integration for JEE Advanced
Frequently Asked Questions
Integration in Maths refers to the process of calculating the area under a curve by breaking it down into smaller sections and adding them together.
The process of finding the antiderivative of a function is known as Integration.
The integral of x is $\int x ,dx = \frac{x^2}{2} + C$
Integral of x = $\int x \; dx = \frac{x^2}{2} + C$, where $C$ is the constant of integration.
The integral of sin x is -cos x + C, where C is an arbitrary constant.
Integral of $\sin x = -\cos x + C$
- Finding the area under a curve
- Computing the volume of a solid of revolution
Integration is used to:
- Find the area under a curve
- Find the velocity of a satellite
- Find the trajectory of a satellite
JEE Study Material (Mathematics)
- 3D Geometry
- Adjoint And Inverse Of A Matrix
- Angle Measurement
- Applications Of Derivatives
- Binomial Theorem
- Circles
- Complex Numbers
- Definite And Indefinite Integration
- Determinants
- Differential Equations
- Differentiation
- Differentiation And Integration Of Determinants
- Ellipse
- Functions And Its Types
- Hyperbola
- Integration
- Inverse Trigonometric Functions
- Limits Continuity And Differentiability
- Logarithm
- Matrices
- Matrix Operations
- Minors And Cofactors
- Properties Of Determinants
- Rank Of A Matrix
- Solving Linear Equations Using Matrix
- Standard Determinants
- Straight Lines
- System Of Linear Equations Using Determinants
- Trigonometry
- Types Of Matrices