Mathematics

Evaluate: $$\displaystyle\int \dfrac { \cos ^ { 3 } x } { \sin ^ { 2 } x + \sin x }$$


SOLUTION
$$I=\displaystyle\int\dfrac{\cos^3x}{\sin^2x+\sin x}dx$$

$$I=\displaystyle\int\dfrac{(\cos^2x).\cos x}{\sin^2x+\sin x}dx$$

$$I=\displaystyle\int\dfrac{(1-\sin^2x).\cos x}{\sin^2x+\sin x}dx$$  

Let $$u=\sin x$$

$$du=\cos xdx$$

$$\displaystyle=\int \dfrac{(1-u^2)\cos x}{u^2+u}.\dfrac{du}{\cos x}$$

$$\displaystyle=\int \dfrac{(1+u)(1-u)}{u(u+1)}du$$

$$\displaystyle=\int \dfrac{1}{u}du-\int \dfrac{u}{u}du$$

$$=ln|u|-u+c$$

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