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Question: If I<sub>n</sub> = \(\int _ { 0 } ^ { 1 } \frac { d x } { \left( 1 + x ^ { 2 } \right) ^ { n , } }\...

If In = ∫01dx(1+x2)n,\int _ { 0 } ^ { 1 } \frac { d x } { \left( 1 + x ^ { 2 } \right) ^ { n , } } nÎN, then which of the following

statement holds good ?

A

2n In + 1 = 2– n + (2n – 1)In

B

I2 = π8+14\frac { \pi } { 8 } + \frac { 1 } { 4 }

C

I2 = π8−14\frac { \pi } { 8 } - \frac { 1 } { 4 }

D

I3 = π16−548\frac { \pi } { 16 } - \frac { 5 } { 48 }

Answer

2n In + 1 = 2– n + (2n – 1)In

Explanation

Solution

In =

= [x(1+x2)]01\left[ \frac { \mathrm { x } } { \left( 1 + \mathrm { x } ^ { 2 } \right) } \right] _ { 0 } ^ { 1 } + ∫2nx2dx(1+x2)n+1\int \frac { 2 n x ^ { 2 } d x } { \left( 1 + x ^ { 2 } \right) ^ { n + 1 } }

= 2–n + 2nIn – 2nIn+1