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REPEATER 2024 — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsREPEATER 20242 Marks
Question
Evaluate: $ \int \frac{e^x(1+x)}{\sin ^2\left(x e^x\right)} d x$
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Answer

$I=\int \frac{e^x(1+x)}{\sin ^2\left(x e^x\right)} d x$
Put $x e^x=t$
Differentiating both sides,
$x \cdot e^x+e^x=\frac{d t}{d x}$
$e^x(1+x) \cdot d x=d t$
$\therefore I=\int \frac{1}{\sin ^2(t)} d t$
$\therefore \quad=\int \operatorname{cosec}^2(t) d t$
$\therefore I=-\cot t+c$$\qquad$$\ldots\ldots\left[\because \int \operatorname{cosec}^2 x \cdot d x=-\cot x+c\right]$
$\therefore \int \frac{e^x(1+x)}{\sin ^2\left(x e^x\right)} d x$$=-\cot \left(x e^x\right)+c$

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