1.

If l_(n)=int_(0)^(pi//4)tan^(n)xdx, n in N then l_(n+2)+l_(n) equals

Answer»

`(1)/(N)`
`(1)/(n-1)`
`(1)/(n+1)`
`(1)/(n+4)`

Solution :`I_(n+1)+I_(n)=int_(0)^(pi//4)tan^(n+2)XDX+int_(0)^(pi//4)tan^(n)xdx`
`rArr I_(n+2)+I_(n)=int_(0)^(pi//4)tan^(n)x(1+tan^(2)x)dx=int_(0)^(pi//4)tan^(n)x.sec^(2)xdx`
`rArr I_(n+2)+I_(n)=int_(0)^(1)t^(n)DT,` where `t=tanx`
`rArr I_(n+2)+I_(n)=(1)/(n+1)`


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