Mathematics

Solve : $$\displaystyle \int \dfrac{\sin x \cos^3 x}{1 + \cos^2 \ x} dx$$


SOLUTION
$$\displaystyle \int \dfrac{\sin x\ \cos^3x}{1+\cos^2x}dx$$

$$\cos x=p$$

$$-\ sinx dx=dp$$

$$\Rightarrow \displaystyle -\int \dfrac{p^3}{1+p^2}dp$$

$$1+p^2=k \Rightarrow p^2=(k-1)$$

$$2pdp=dk$$

$$\Rightarrow \displaystyle \int \dfrac{(k-1)dk}{2(k)}$$

$$\therefore \displaystyle \int \dfrac{1}{2}dx+\dfrac{1}{2} \int \dfrac{dx}{k}$$

$$\therefore \displaystyle \int -\dfrac{k}{2}+\dfrac{1}{2}ln|k|$$

$$\therefore  \dfrac{-(1+p^2)}{2}+\dfrac{1}{2}ln|1+p^2|$$

$$\therefore \boxed{\dfrac{1}{2}\left[ln|1+\cos^2x)-(1+\cos^2x)\right]}$$
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Subjective Medium Published on 17th 09, 2020
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