# ∫ E X ( 1 + Sin X 1 + Cos X ) D X - Mathematics

Sum
$\int e^x \left( \frac{1 + \sin x}{1 + \cos x} \right) dx$

#### Solution

$\text{ Let I } = \int e^x \left( \frac{1 + \sin x}{1 + \cos x} \right) dx$

$= \int e^x \left( \frac{1}{1 + \cos x} + \frac{\sin x}{1 + \cos x} \right) dx$

$= \int e^x \left( \frac{1}{2 \cos^2 \frac{x}{2}} + \frac{2 \sin \frac{x}{2} \cos \frac{x}{2}}{2 \cos^2 \frac{x}{2}} \right) dx$

$= \int e^x \left( \frac{1}{2} \sec^2 \frac{x}{2} + \tan \frac{x}{2} \right) dx$

$\text{ Putting e}^x \tan \frac{x}{2} = t$

$\text{ Diff both sides w . r . t . x }$

$e^x \cdot \tan \left( \frac{x}{2} \right) + e^x \times \frac{1}{2} \sec^2 \frac{x}{2} = \frac{dt}{dx}$

$\Rightarrow e^x \left[ \tan \frac{x}{2} + \frac{1}{2} \sec^2 \left( \frac{x}{2} \right) \right] dx = dt$

$\therefore \int e^x \left( \frac{1}{2} \sec^2 \frac{x}{2} + \tan \frac{x}{2} \right) dx = \int dt$

$= t + C$

$= e^x \tan\left( \frac{x}{2} \right) + C$

Concept: Indefinite Integral Problems
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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 19 Indefinite Integrals
Exercise 19.26 | Q 3 | Page 143