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Let I(n)=int(0)^(pi//2)(sinx+cosx)^(n)dx...

Let `I_(n)=int_(0)^(pi//2)(sinx+cosx)^(n)dx(nge2)`. Then the value of n. `I_(n)-2(n-1)I_(n-1)`is

A

5

B

9

C

2

D

7

Text Solution

Verified by Experts

The correct Answer is:
C
`I_(n)=int_(0)^(pi//2)(sinx+cosx)^(n-1)(sinx+cosx)dx`
`" "=[(sinx+cosx)^(n-1)(sinx-cosx)]_(0)^(pi//2)`
`" "+int_(0)^(pi//2)(sinx+cosx)^(n-2)(cosx-sinx)^(2)dx`
`n.I_(n)=2+2(n-1)I_(n-2)rArrn.I_(n)-0 2(n-1)I_(n-2)=2`
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