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Manipal MET2014MathematicsDefinite Integration

The value of _ 1 / n ^ (a n-1) / n x a-x + x d x is equal to

Options

  1. Aa 2
  2. Bn a+2 2 n
  3. Cn a-2 2 n
  4. Dn a 2

Correct answer

C. n a-2 2 n

Step-by-step solution

aligned & Let, I= _ 1 n ^ (a n-1) / n x a-x + x d x & = _ 1 n ^ a- 1 n x a-x + x d x ...(i) & = _ 1 n ^ a- 1 n ( 1 n +a- 1 n -x ) a- ( 1 n +a- 1 n -x ) + 1 n +a- 1 n -x d x & I= _ 1 n ^ a- 1 n a-x x + a-x d x...(ii) aligned On adding Eqs. (i) and (ii), we get aligned 2 I & = _ 1 n ^ a- 1 n 1 d x=[x]_ 1 n ^ a- 1 n & =a- 1 n - 1 n = n a-2 n I= n a-2 2 n aligned

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