∫ cos x / [(2 + sin x)(4 + sin x)] dx

 find ∫ cos x / [(2 + sin x)(4 + sin x)] dx 

integration by substitution and partial fractions

Substitution

Let t = sin x

Then dt = cos x dx

 ∫ cos x / [(2 + sin x)(4 + sin x)] dx =  ∫ 1 / [(2 + t)(4 + t)] dt

Method of partial fractions

 1 / [(t + 2(t + 4)] = A/(t + 2) + B/(t + 4)

multiply [t+2][t+4]

1 = A(t + 4) + B(t + 2)

Put t = −2

1 = A(2) 

⇒ A = ½

Put t = −4:

1 = B(−2) 

⇒ B = −½

1 / [(t + 2)(t + 4)] = ½[1/(t + 2) − 1/(t + 4)]

integrate

I = ½  ∫ [1/(t + 2)  − 1/(t + 4)] dt 

= ½[ln|t + 2| − ln|t + 4|] + C

= ½ ln|(t + 2)/(t + 4)| + C   use  t = sin x

=½ ln[(2 + sin x)/(4 + sin x)] + C

for  explanation watch this video