Mathematics

The value of the integration $$\displaystyle\int^{\pi/4}_{-\pi/4}\left(\lambda |\sin x|+\dfrac{\mu \sin x}{1+\cos x}+\gamma\right)dx$$.


ANSWER

Is independent of $$\mu$$ only


SOLUTION
$$I=2\lambda\displaystyle\int^{\dfrac{\pi}{4}}_0\sin x dx+\mu\displaystyle\int^{\dfrac{\pi}{4}}_{-\dfrac{\pi}{4}}\tan \dfrac{x}{2}dx+\gamma\displaystyle\int^{\dfrac{\pi}{4}}_{-\dfrac{\pi}{4}}dx$$
$$=2\lambda\left(1-\dfrac{1}{\sqrt{2}}\right)-0+\gamma\left(\dfrac{\pi}{2}\right)$$
$$(\because \tan\dfrac{x}{2}$$ is an odd function$$)$$.
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Single Correct Medium Published on 17th 09, 2020
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