∫ (cos x)/((2 + sin x)(1 + sin x))dx, x ∈ [0, π/2] equals A. log(2/3) B. log(3/2) C. log(3/4) D. log(4/3)

Question

\(\int\limits_{0}^{\pi/2}\cfrac{cos\text x}{(2+sin\,\text x)(1+sin\,\text x)}d\text x \) equals

A. log(\(\cfrac23\))

B. log(\(\cfrac32\))

C. log(\(\cfrac34\))

D. log(\(\cfrac43\))

Answer 1

Let, sin x = t

Differentiating both side with respect to x

\(\cfrac{dt}{dx}=cos\,x\)

⇒dt = cos x dx

At x = 0, t = 0

At x = π\2, t = 1

By using the concept of partial fraction

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

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

A + 2B = 1, A + B = 0

A = -1, B = 1