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KCET2026MathematicsIndefinite Integration

If ' n ' is a natural number, then ^n x ^ n+2 x ,dx =

Options

  1. A^ n-1 x n - 1 + C
  2. B^n x n + C
  3. C^ n+2 x n + 2 + C
  4. D^ n+1 x n + 1 + C

Correct answer

D. ^ n+1 x n + 1 + C

Step-by-step solution

The given integral is I = ^n x ^ n+2 x ,dx This can be rewritten as: I = ( x x )^n 1 ^2 x ,dx I = ^n x ^2 x ,dx Let x = t , then ^2 x ,dx = dt Substituting these into the integral, we get: I = t^n ,dt I = t^ n+1 n+1 + C I = ^ n+1 x n+1 + C Answer: ^ n+1 x n + 1 + C

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