∫ sin^(13) x cos^(12) xdx x ∈ to (-1, 1) =

Question

\(\int\limits_{-1}^1 \sin ^{13} x \cos ^{12} x d x=\)

(A) 0

(B) 1

(C) \(\frac{1}{2}\)

(D) 2

Answer 1

Correct option is (A) 0

\(I =\int\limits_{-1}^1 \sin ^{13} x \cos ^{12} x \ dx \)

The integrand \(f(x) = \sin^{13} (x) \cos ^{12} (x)\) is an odd function because

\(f(-x) = (-\sin (x))^{13} (\cos (-x))^{12} = - f(x)\)

The integral of an odd function over a symmetric limit (-1, 1) is always zero.

\(\therefore \int\limits_{-1}^1 \sin ^{13} x \cos ^{12} x \ dx = 0\)

Answer 2

Correct option is (A) 0