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If l(n)=int(0)^(pi//4) tan^(n)x dx, n in...

If `l_(n)=int_(0)^(pi//4) tan^(n)x dx, n in N "then" I_(n+2)+I_(n)` equals

A

`(1)/(n)`

B

`(1)/(n-1)`

C

`(1)/(n+1)`

D

`(1)/(n+2)`

Text Solution

Verified by Experts

The correct Answer is:
C
We have,
`I_(n+2)+I_(n)=underset(0)overset(pi//4)int tan^(n+2) x dx+underset(0)overset(pi//4)int tan^(n) x dx`
`rArr I_(n+2)+I_(n)=underset(0)overset(pi//4)int tan^(n)(1+tan^(2)x) dx+underset(0)overset(pi//4)int tan^(n) x sec^(2)x dx`
`rArr I_(n+2)+I_(n)=underset(0)overset(pi//4)int t^(n)dt,` where, `t= tan x`
`rArr I_(n+2)+I_(n)(1)/(n+1)`
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