Mathematics

Evaluate$$\int\limits_{} {\dfrac{{\cos x}}{{\left( {2 + \sin x} \right)\left( {3 + 4\sin x} \right)}}dx.} $$


SOLUTION
$$\displaystyle \int \dfrac{\cos x\,\,\, dx}{(2+\sin x)(3+4sin x)}$$

$$\sin x=t$$

$$\cos dx=dt$$
$$\displaystyle \int \dfrac{dt}{(2+t)(3+4t)}$$

Let $$\dfrac{1}{(2+t)(3+4t)}=\dfrac{A}{2+t}+\dfrac{B}{3+4t}$$

$$I=A(3+4t)+B(2+t)$$
$$3A+2B=1$$
$$4A+B=0\,\,\,\,       \Rightarrow B=-4A$$
$$3A-8A=1$$
$$A=-\dfrac{1}{5}$$            $$B=\dfrac{4}{5}$$

$$=\displaystyle I=\int \dfrac{dt}{(2+t)(3+4t)}=\dfrac{-1}{5}\int \dfrac{dt}{2+t}+\dfrac{4}{5}\int \dfrac{dt}{3+4t}$$
$$=\dfrac{-1}{5}\log |2+t|+\dfrac{4}{5}\dfrac{\log |3+4t|}{4}+c$$

$$=\dfrac{-1}{5}\log |2+\sin x|+\dfrac{1}{5}\log |3+4\sin x|+c$$

$$=\dfrac{1}{5}\log |\dfrac{3+4\sin x}{2+\sin x |}+c$$
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