If `int_0^oosinx/xdx=pi/2`, then `int_0^oosin^3x/xdx` is equal to

Options

A. `pi//2`
B. `pi//4`
C. `pi//6`
D. `3pi//2`

Answer

B

Explanation

`sin^(3)x=3/4 sinxx =1/4 sinx 3x`
`:. int_(0)^(oo) (sin^(3)x)/x dx=3/4 int_(0)^(oo) (sinx)/x dx-1/4 int_(0)^(oo) (sin 3x)/x dx`
`=3/4 int_(0)^(oo) (sinx)/x dx-1/4 int_(o)^(oo) (sinu)/u du (u=3x)`
`=3/4 (pi)/2-1/4 (pi)/2=(pi)/4`