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Question Number 22623 by tawa tawa last updated on 21/Oct/17

How is : \int \frac{x + \sin (x)}{1 + \cos (x)} dx = xtan (\frac{x}{2}) + C

Answered by vajpaithegrate@gmail.com last updated on 21/Oct/17

\int \frac{x + 2 \sin \frac{x}{2} \cos \frac{x}{2}}{2 \cos^{2} \frac{x}{2}} dx

\int \frac{x}{{2 \cos}^{2} \frac{x}{2}} + \frac{2 \sin \frac{x}{2} \cos \frac{x}{2}}{2 \cos^{2} \frac{x}{2}}

= \int \frac{x}{2} \sec^{2} \frac{x}{2} + \tan \frac{x}{2} dx

= \int d (xtan (\frac{x}{2})) dx

= xtan \frac{x}{2} + c

[\sin x = 2 \sin \frac{x}{2} \cos \frac{x}{2}; 1 + \cos x = 2 \cos^{2} \frac{x}{2}]

Commented by tawa tawa last updated on 21/Oct/17

God bless you sir