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Fundamental Formulae of Indefinite Integration (Inverse Trigonometric Functions) - Practice Questions & MCQ

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

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  • 9 Questions around this concept.

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Evaluate the integral of \int\frac{1}{5\sqrt{25-4x^{2}}}dx.

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Concepts Covered - 1

Fundamental Formulae of Indefinite Integration (Inverse Trigonometric Functions)

Fundamental Formulae (Inverse Trigonometric Functions)

\\\mathrm{1.\;\;{\frac{d}{d x}\left(\sin ^{-1} \frac{x}{a}\right)=\frac{1}{\sqrt{a^{2}-x^{2}}}}\;\;\; {\Rightarrow \int \frac{d x}{\sqrt{a^{2}-x^{2}}}=\sin ^{-1}\left(\frac{x}{a}\right)+C}}\\\\\\\mathrm{2.\;\;{\frac{d}{d x}\left(\cos ^{-1} \frac{x}{a}\right)=\frac{-1}{\sqrt{a^{2}-x^{2}}}} \;\; {\Rightarrow \int \frac{-1}{\sqrt{a^{2}-x^{2}}} d x=\cos ^{-1}\left(\frac{x}{a}\right)+C}}\\\\\\\mathrm{3.\;\;{\frac{d}{d x}\left(\frac{1}{a} \tan ^{-1} \frac{x}{a}\right)=\frac{1}{a^{2}+x^{2}}} \;\; {\Rightarrow \int \frac{d x}{a^{2}+x^{2}}=\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)+C}}

\\\\\\\mathrm{4.\;\;{\frac{d}{d x}\left(\frac{1}{a} \cot ^{-1} \frac{x}{a}\right)=\frac{-1}{a^{2}+x^{2}}} \;\; {\Rightarrow \int \frac{-1}{a^{2}+x^{2}} d x=\frac{1}{a} \cot ^{-1}\left(\frac{x}{a}\right)+C}}\\\\\\\mathrm{5.\;\;{\frac{d}{d x}\left(\frac{1}{a} \sec ^{-1} \frac{x}{a}\right)=\frac{1}{x \sqrt{x^{2}-a^{2}}}} \;\; {\Rightarrow \int \frac{d x}{x \sqrt{x^{2}-a^{2}}}=\frac{1}{a} \sec ^{-1}\left(\frac{x}{a}\right)+C}}\\\\\\\mathrm{6.\;\;{\frac{d}{d x}\left(\frac{1}{a} \csc ^{-1} \frac{x}{a}\right)=\frac{-1}{x \sqrt{x^{2}-a^{2}}}} \;\; {\Rightarrow \int \frac{-d x}{x \sqrt{x^{2}-a^{2}}}=\frac{1}{a} \csc ^{-1}\left(\frac{x}{a}\right)+C}}

 

You can derive all the above results using substitution method.

For example, to get result (1), substitute  x = a sin Ө or x = a cos Ө and solve 

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Fundamental Formulae of Indefinite Integration (Inverse Trigonometric Functions)

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Fundamental Formulae of Indefinite Integration (Inverse Trigonometric Functions)

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 7.17

Line : 1

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