Mathematics

Evaluate: $$\displaystyle\int \dfrac { \cos x + x \sin x  } { x ( x + \cos x ) } d x $$


SOLUTION
$$I=\displaystyle\int \dfrac{\cos x+x\sin x}{x(x+\cos x)}dx$$

$$I=\displaystyle\int \dfrac{(\cos x+x)+(x\sin x-x)}{x(x+\cos x)}dx$$

$$I=\displaystyle\int \dfrac{1}{x}\dfrac{(x+\cos x)}{(x+\cos x)}+\displaystyle\int \dfrac{x(\sin x-1)}{x(x+\cos x)}dx$$

$$I=\displaystyle\int \dfrac{1}{x}dx+\displaystyle\int \dfrac{(\sin x-1)}{x+\cos x}dx$$

Let $$x+\cos x=u$$

$$du=(1-\sin x)dx$$

$$I=ln|x|+\displaystyle\int \dfrac{-(1-\sin x)}{u}\dfrac{du}{(1-\sin x)}$$

$$I=\ln|x|-\ln|u|+c$$

$$I=\ln|x|-\ln|x+\cos x|+c$$.
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Subjective Medium Published on 17th 09, 2020
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