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AP EAMCET202421 May 2024Morning ShiftMathematicsDefinite IntegrationActual

If n 2 is a natural number and 0 2 , then ( ^n - )^ 1 / n ^ n+1 d =

Options

  1. An n-1 ( ^ (1-n) -1 )^2+c
  2. Bn (n+1)(1-n) ( ^ (1-n) -1 )^ 1+ 1 n +c
  3. Cn 1-n ( ^ (n-1) -1 )^2+c
  4. Dn 1-n^2 (1- ^ (n-1) )^ (n+1) / n

Correct answer

D. n 1-n^2 (1- ^ (n-1) )^ (n+1) / n

Step-by-step solution

I = ( ^n - )^ 1 n ^ n+1 d = (1- ^ 1-n )^ 1 n ^ n+1 d = (1- ^ 1-n )^ 1 n ^n d Let 1- ^ 1-n =' t aligned & (1- n ) ^ - n ~d = dt & I = t^ 1 n (1-n) d t= 1 t^n+1 ( 1 n +1 )(1-n) = n t^ 1+n n (1-n^2 ) & = n 1-n^2 (1- ^ (1-n) )^ n+1 n aligned

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