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Property 7

∫ x+1 exsin 2...

Question

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

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Solution

$\int \frac{(x + 1) e^{x}}{\sin^{2} (x e^{x})} d x Let x e^{x} = t \Rightarrow (1 \cdot e^{x} + x e^{x}) = \frac{d t}{d x} \Rightarrow (x + 1) e^{x} d x = d t Now, \int \frac{(x + 1) e^{x}}{\sin^{2} (x e^{x})} d x = \int \frac{d t}{\sin^{2} t} = \int {cosec}^{2} t d t = - \cot (t) + C = - \cot (x e^{x}) + C$

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