int frac{e^x(1+x)}{sin ^2(x cdot e^x)} dx = | vedclass

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Question

(A) Let $I = \int \frac{e^x(1+x)}{\sin^2(x \cdot e^x)} dx$. Substitute $u = x \cdot e^x$. Then,$du = (1 \cdot e^x + x \cdot e^x) dx = e^x(1+x) dx$. Su

Vedclass · Accepted Answer

$-\cot(x \cdot e^x)$

Vedclass · Answer

(A) Let $I = \int \frac{e^x(1+x)}{\sin^2(x \cdot e^x)} dx$. Substitute $u = x \cdot e^x$. Then,$du = (1 \cdot e^x + x \cdot e^x) dx = e^x(1+x) dx$. Substituting these into the integral,we get: $I = \int \frac{1}{\sin^2(u)} du = \int \csc^2(u) du$. Since $\int \csc^2(u) du = -\cot(u) + C$,we have $I = -\cot(x \cdot e^x) + C$. Thus,the correct option is $A$.

$\int \frac{e^x(1+x)}{\sin ^2(x \cdot e^x)} dx = $ . . . . . . $+ C$.

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