If `I_n=int_0^pi e^x(sinx)^n dx ,` then `(I_3)/(I_1)` is equal to

Monjur Alahi

If `I_n=int_0^pi e^x(sinx)^n dx ,` then `(I_3)/(I_1)` is equal to
A. `3//5`
B. `1//5`
C. `1`
D. `2//5`


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Salim Ahmed Kawsar

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Correct Answer - A
`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`

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