Mathematics

$$\int \dfrac{\cos x}{(1+\sin x)(2+\sin x)}dx$$


SOLUTION
We have,
$$\int \dfrac{\cos x}{(1+\sin x)(2+\sin x)}dx$$

Let $$y=1+\sin x$$
$$dy=\cos x.dx$$

Therefore,
$$\Rightarrow\int\dfrac{dy}{y(1+y)}=\int\dfrac{(1+y)-y}{y(1+y)}dy$$
$$\Rightarrow\int\left(\dfrac{1}{y}-\dfrac{1}{1+y}\right)dy$$
$$\Rightarrow \ln (y)-\ln (1+y)+c$$
$$\Rightarrow \ln(1+\sin x)-\ln(2+\sin x)+c$$

Hence, this is the answer.
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