`(626int_(0)^(oo)e^(-x)sin^(25)xdx)/(int_(0)^(oo)e^(-x)sin^(23)xdx)` is equal to

Options

A. 300
B. 625
C. 600
D. 1200

Answer

C

Explanation

Let
`I_(25)=int_(0)^(oo)e^(-x)sin^(25)xdx`
`=(-e^(-x)sin^(25)x)|_(0)^(oo)+25int_(0)^(oo)e^(-x)sin^(24)x.cos xdx`
`=0+25int_(0)^(oo)e^(-x)sin^(24)x.cosxdx`
`=(-e^(-x)sin^(24)x cosx)|_(0)^(oo)+25int_(0)^(oo)e^(-x)(24sin^(23)xcos^(2)x-sin^(25)x)dx`
`=0+25int_(0)^(oo)e^(-x)(24sin^(23)x-25sin^(25)x)dx`
`=600I_(23)-625I_(25)`
`therefore" "626(I_(25))/(I_(23))=600`