Mathematics

Evaluate $$\displaystyle\int^{\pi}_0\dfrac{x}{1+\sin x}dx$$.


ANSWER

$$x\tan x + ln|\cos x| - x\sec x -ln|\sec x - \tan x| + C$$


SOLUTION
$$\displaystyle \int \cfrac{x(1-\sin x)}{1-\sin^2x}dx$$
$$\displaystyle \int \cfrac{x(1-\sin x)}{\cos^2x}dx$$
$$\displaystyle \int (x \sec^2x - x \tan x \sec x )dx$$
Applying byparts,we get
$$x\tan x + ln|\cos x| - x\sec x -ln|\sec x - \tan x| + C$$
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Single Correct Medium Published on 17th 09, 2020
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