Rajasthan Boardहिन्दी माध्यमकक्षा 12 साइन्सगणितसमाकलन1 Mark
Question
$\int \frac{e^{x}(1+x)}{\cos ^{2}\left(e^{x} x\right)} d x$ बराबर है:
✓
Answer
माना $x e^{x}=t$
$\Rightarrow\left(x e^{x}+e^{x}\right) =\frac{d t}{d x}$
$\Rightarrow d x=\frac{d t}{e^{x}(x+1)}$
$\therefore \int \frac{e^{x}(1+x)}{\cos ^{2}\left(e^{x} x\right)} d x =\int \frac{e^{x}(1+x)}{\cos ^{2} t} \times \frac{d t}{e^{x}(1+x)}$
$=\int \frac{1}{\cos ^{2} t} d t = \in t \sec^2 t \ dt = \tan t + C = \tan (xe^x) + C$
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