int frac{e^{x}(1+x)}{cos ^{2}(x e^{x})} d x e | vedclass

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Question

(D) Let $I = \int \frac{e^{x}(1+x)}{\cos ^{2}(x e^{x})} d x$. Substitute $t = x e^{x}$. Differentiating both sides with respect to $x$,we get: $dt = (

Vedclass · Accepted Answer

$	an (x e^{x})+C$

Vedclass · Answer

(D) Let $I = \int \frac{e^{x}(1+x)}{\cos ^{2}(x e^{x})} d x$. Substitute $t = x e^{x}$. Differentiating both sides with respect to $x$,we get: $dt = (e^{x} + x e^{x}) dx = e^{x}(1+x) dx$. Substituting these into the integral: $I = \int \frac{dt}{\cos ^{2} t} = \int \sec ^{2} t dt$. Integrating $\sec ^{2} t$,we get: $I = 	an t + C$. Substituting back $t = x e^{x}$: $I = 	an (x e^{x}) + C$. Thus,the correct option is $D$.

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

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