Match List – I with List – II.
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LIST I |
LIST II |
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A. $\int x^2 e^{x^3} d x$ |
I. $\frac{1}{2 a} \log \left|\frac{a+x}{a-x}\right|+ C$ + C is an arbitrary constant |
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B. $\int e^x\left(\tan ^{-1} x+\frac{1}{1+x^2}\right) d x$ |
II. $\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)+ C$ + C is an arbitrary constant |
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C. $\int \frac{d x}{a^2-x^2}$ |
III. $\frac{1}{3} e^{x^3}+ C$ + C is an arbitrary constant |
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D. $\int \frac{d x}{x^2+a^2}$ |
IV. $e^x \tan ^{-1} x+ C$ + C is an arbitrary constant |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A-III, B-IV, C-I, D-II
(A) $\frac{3}{3}\int x^2 e^{x^3} d x=\frac{e^{x^3}}{3}+ C$ (III)
(B) $\int e^x\left(\underset{f(x)}{\underbrace{\tan ^{-1} x}}+\underset{f'(x)}{\underbrace{\frac{1}{1+x^2}}}\right) d x=e^x \tan ^{-1} x+ C$ (IV)
(C) $\int \frac{d x}{a^2-x^2}=\frac{1}{2a}\int\left(\frac{1}{a-x}+\frac{1}{a+x}\right)dx$
$=\frac{1}{2a}(\log|a+x|-\log|a-x|)=\frac{1}{2a}\log|\frac{a+x}{a-x}|+C$ (I)
(D) $\int \frac{d x}{x^2+a^2}=\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)+ C$ (II)