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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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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