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Question: If I<sub>n</sub> = \(\int_{0}^{1}\frac{dx}{(1 + x^{2})^{n,}}\)nÎN, then which of the following state...

If In = ∫01dx(1+x2)n,\int_{0}^{1}\frac{dx}{(1 + x^{2})^{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 =

=+ ∫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