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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
Question
Integrate the function $\frac{2+\sin 2 x}{1+\cos 2 x} e^{x}$
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Answer

Let $I=\frac{2+\sin 2 x}{1+\cos 2 x} e^{x}$ 
= $\left(\frac{2+2 \sin x \cos x}{2 \cos ^{2} x}\right) e^{x}$ 
= $2 \cdot\left(\frac{1+\sin x \cos x}{2 \cos ^{2} x}\right) e^{x}$ 
= $\left(\frac{1}{\cos ^{2} x}+\frac{\sin x \cos x}{\cos ^{2} x}\right) e^{x}$ 
= $\left(\sec ^{2} x+\tan x\right) e^{x}$ 
$\Rightarrow \int \frac{2+\sin 2 x}{1+\cos 2 x} e^{x} d x=\int\left(\sec ^{2} x+\tan x\right) e^{x} d x$ 
Now let tan x = f(x)
$\Rightarrow$ f'(x) = sec2x dx 
$\Rightarrow \int\left(\sec ^{2} x+\tan x\right) e^{x} d x = \int\left(f(x)+f^{\prime}(x)\right) e^{x} d x = e^{x} f(x)+C$
$\Rightarrow I=e^{x} \tan x+C$

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