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Integrals of Particular Function - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Some Special Integration, Application of Special Integral Formula (Part 1) is considered one of the most asked concept.

  • 27 Questions around this concept.

Solve by difficulty

QuestionMEDIUM

  Evaluate the integral of \int \frac{dx}{\sqrt{x^{2}+49}}:

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Find the integral of\int \frac{dx}{9-x^{2}}

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The integral value of \int\frac{x\ dx}{\sqrt{x^{4}+1}}

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Concepts Covered - 3

Some Special Integration

\\\mathrm{1.\;\;\int \frac{d x}{x^{2}+a^{2}}=\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)+C}\\\\\mathrm{\;\;\;\;\;\;put\;\;x=a\tan\theta,\;then\;\;dx=a\sec^2\theta\;d\theta}\\\mathrm{\;\;\;\;\;\;\therefore \;\;\int \frac{d x}{x^{2}+a^{2}}=\int\frac{a\sec^2\theta}{a^2+a^2\tan^2\theta}\;d\theta}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\int\frac{a\sec^2\theta}{a^2\left (1+\tan^2\theta \right )}\;d\theta}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\frac{1}{a}\int d\theta=\frac{1}{a}\theta+C=\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)+C}

 

\\\mathrm{2.\;\;\int \frac{d x}{x^{2}-a^{2}}=\frac{1}{2a} \log\left | \frac{x-a}{x+a} \right |+C}\\\\\text{\;\;\;\;\;\;\;we can rewrite above integral as}\\\mathrm{\;\;\;\;\;\;\int \frac{d x}{x^{2}-a^{2}}=\frac{1}{2a}\int\left ( \frac{1}{x-a}-\frac{1}{x+a} \right )dx}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\frac{1}{2a}\left ( \log|x-a|-\log|x+a| \right )+c}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\frac{1}{2a} \log\left | \frac{x-a}{x+a} \right |+C}

 

\\\mathrm{3.\;\;\int \frac{d x}{a^{2}-x^{2}}=\frac{1}{2 a} \log \left|\frac{a+x}{a-x}\right|+C}\\\\\mathrm{4.\;\;\int \frac{d x}{\sqrt{a^{2}-x^{2}}}=\sin ^{-1}\left(\frac{x}{a}\right)+C}\\\\\mathrm{5.\;\;\int \frac{d x}{\sqrt{a^{2}+x^{2}}}=\log |x+\sqrt{x^{2}+a^{2}}|+C}\\\\\mathrm{6.\;\;\int \frac{d x}{\sqrt{x^{2}-a^{2}}}=\log |x+\sqrt{x^{2}-a^{2}}|+C}

\\\\\mathrm{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}\\\\\mathrm{8.\;\;\int \sqrt{a^{2}+x^{2}} d x =\frac{1}{2} x \sqrt{a^{2}+x^{2}}+\frac{1}{2} a^{2} \log |x+\sqrt{a^{2}+x^{2}}|+C}\\\\\mathrm{9.\;\;\int \sqrt{x^{2}-a^{2}} d x=\frac{1}{2} x \sqrt{x^{2}-a^{2}}-\frac{1}{2} a^{2} \log |x+\sqrt{x^{2}-a^{2}}|+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}

Application of Special Integral Formula (Part 1)

Integration of the type 

\\ {\text { (i) } \int \frac{(p x+q) \,dx}{a x^{2}+b x+c} }\\\\ \text { (ii) } \int \frac{(p x+q)}{\sqrt{a x^{2}+b x+c}} \,dx\\\\ {\text { (iii) } \int(p x+q) \sqrt{a x^{2}+b x+c} \,\,dx}\\

\\\mathrm{Express\;the\;linear\;factor\;\mathit{px+q}\;in\;terms\;of\;the\;derivative\;of}\\\mathrm{quadratic\;factor\;\mathit{ax^2+bx+c}}\\\text{i.e.}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;{p x+q=\lambda.\frac{ d}{d x}\left(a x^{2}+b x+c\right)+\mu}}\\\\\mathrm{\Rightarrow \;\;\;\;\;\;\;\;{p x+q=\lambda(2 a x+b)+\mu}}\\\\\mathrm{{\text {Find } \lambda \text { and } \mu \text { and replace }(p x+q) \text { by }} \\ {\lambda(2 a x+b)+\mu}}

Application of Special Integral Formula (Part 2)

Integration of the type

\\\mathrm{1.\;\;\int \frac{a x^{2}+b x+c}{\left(p x^{2}+q x+r\right)} \,\,d x}\\\\\mathrm{2.\;\;\int \frac{\left(a x^{2}+b x+c\right)}{\sqrt{p x^{2}+q x+r}} \,\,d x}\\\\\mathrm{3.\;\;\int\left(a x^{2}+b x+c\right) \sqrt{p x^{2}+q x+r} \,\,dx}

\\\text{Substitute,}\\\\\mathrm{a x^{2}+b x+c=\lambda\left(p x^{2}+q x+r\right)+\mu\left\{\frac{d}{d x}\left(p x^{2}+q x+r\right)\right\}+\gamma}

Find λ, μ and γ. These integrations reduces to integration of three independent functions.

 

Integration of the form \mathbf{\int \frac{k(x)}{ax^2+bx+c}\;dx}

(here, k(x) is a polynomial of degree greater than 2)

To solve this type of integration, divide the numerator by the denominator and express the intagral as 

\mathrm{Q(x)+\frac{R(x)}{ax^2+bx+c}} 

Here, R(x) is a linear function of x.

Study it with Videos

Some Special Integration
Application of Special Integral Formula (Part 1)
Application of Special Integral Formula (Part 2)

Study other Related Concepts

Integrals of Particular Function Current Topic

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Integrals of Particular Function
Trigonometric Integrals
Integration By Parts Formula
Integration Using Partial Fraction
Integration of Irrational Algebraic Function
Reduction Formula

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Books

Reference Books

Some Special Integration

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 7.14

Line : 1

Application of Special Integral Formula (Part 1)

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 7.19

Line : 1

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