The value of `int underset(0)overset(pi) xsin^(3)xdx` is

Options

A. `(4pi)/(3)`
B. `(2pi)/(3)`
C. 0
D. None of these

Answer

B

Explanation

Let `I= int _(0)^(pi)x sin^(3) dx " " ....(i)`
Also, `I= int_(0)^(pi)(pi-x)sin^(3)x dx " "...(ii)`
On adding Eqs(i) and (ii), we get
`2I= pi int_(0)^(pi) sin^(3) x dx`
`=(pi)/(4)int_(0)^(pi) (3 sin x- sin 3x)dx`
`=(pi)/(4)[- 3 cos x+( cos 3x)/(3)] _(0)^(pi)`
`=(pi)/(4)[ 3-(1)/(3)+3-(1)/(3)]=(4pi)/(3)`
Hence, `I=(2pi)/(3)`