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)
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
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