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JEE Advanced2007MathematicsIndefinite IntegrationActual

Let f(x)= x (1+x^n )^ 1 / n for n 2 and g(x)= (f f o o f) _ f occurs n times (x) . Then, x^ n-2 g(x) d x equals

Options

  1. A1 n(n-1) (1+n x^n )^ 1- 1 n +k
  2. B1 n-1 (1+n x^n )^ 1- 1 n +k
  3. C1 n(n+1) (1+n x^n )^ 1+ 1 n +k
  4. D1 n+1 (1+n x^n )^ 1+ 1 n +k

Correct answer

A. 1 n(n-1) (1+n x^n )^ 1- 1 n +k

Step-by-step solution

Here, array rlrl & and & f f(x) & = f(x) [1+f(x)^n ]^ 1 / n = x (1+2 x^n )^ 1 / n & f f f(x) & = x (1+3 x^n )^ 1 / n & Let & g(x) & = (f f o o f) _ n times (x)= x (1+n x^n )^ 1 / n I & = x^ n-2 g(x) d x= x^ n-1 d x (1+n x^n )^ 1 / n & & = 1 n^2 n^2 x^ n-1 d x (1+n x^n )^ 1 / n = 1 n^2 d d x (1+n x^n ) (1+n x^n )^ 1 / n d x array I= 1 n(n-1) (1+n x^n )^ 1- 1 n +k .

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