`IfI_n=int_0^pie^x(sin"x")^("n")dx ,t h e n(I_3)/(I_1)"is equal to"` (a) `3/5` (b) `1/5` (c) 1 (d) `2/5`

Options

A. `3//5`
B. `1//5`
C. `1`
D. `2//5`

Answer

A

Explanation

`I_(3)=int_(0)^(pi)e^(x)(sinx)^(3)dx`
`=e^(x)(sinx)^(3)|_(0)^(pi)-3int_(0)^(pi)(sinx)^(2)cosx e^(x)dx`
`=0-3(sinx)^(2)cosx e^(x)|_(0)^(pi)+3int_(0)^(pi)(2sinx cos x cosx)`
`-sin x sin^(2)x)e^(x)dx`
`=0+6int_(0)^(pi)sin x cos^(2) xe^(2) dx-3 int_(0)^(pi) sin^(3) xe^(x) dx`
`=6int_(0)^(pi) sinx(1-sin^(2)x)e^(x)dx-3int_(0)^(pi)sin^(3)xe^(x)dx`
`=6int_(0)^(pi) sinxe^(x)dx-9int_(0)^(pi) sin^(3)x e^(x)dx=6I_(1)-9I_(3)`
or `10 I_(3)=6I_(1)`
or `(I_(3))/(I_(1))=3/5`