The correct option is A tan (xe^x) + C Let I=∫e^x(x+1)/cos^2(xe^x)dx =∫ e^x(x+1)sec^2(xe^x)dx Put xe^xdx=t ⇒ e^x (x+1)dx=dt I= ∫^ ...

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Integration by Substitution

∫exx+1/cos2xe...

Question

$\int \frac{e^{x} (x + 1)}{c o s^{2} (x e^{x})} d x =$

t a n (x e^{x}) + C

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s e c (x e^{x}) t a n (x e^{x}) + C

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- t a n (x e^{x}) + C

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None of these

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Solution

The correct option is A $t a n (x e^{x}) + C$
Let $I = \int \frac{e^{x} (x + 1)}{c o s^{2} (x e^{x})} d x$
$= \int e^{x} (x + 1) s e c^{2} (x e^{x}) d x$
Put $x e^{x} d x = t$
$\Rightarrow e^{x} (x + 1) d x = d t$
$I = \int {sec}^{2} t d t = tan t + C$
$I = tan (x e^{x}) + C$

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Similar questions

\int \frac{e^{x} (1 + x)}{\cos^{2} (x e^{x})} d x =

equals

A. − cot (ex^x) + C

B. tan (xe^x) + C

C. tan (e^x) + C

D. cot (e^x) + C

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\int \frac{(x + 1)}{x (1 + x e^{x})^{2}} d x

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