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If I n = ∫ x n e 6 x d x , then the expression 6 I 10 + 10 I 9 simplifies to (where, c is the constant of integration)

Options

  1. Ax 10 e 5 x + c
  2. Bx 10 e 6 x + c
  3. Cx 9 e 5 x + c
  4. Dx 10 e 10 x + c

Correct answer

B. x 10 e 6 x + c

Step-by-step solution

Using integration by parts in I n , we get, I n = ∫ x n ⏟ I e 6 x ⏟ I I d x = x n e 6 x 6 - ∫ n x n - 1 e 6 x 6 d x ⇒ 6 I n = x n e 6 x - n I n - 1 + c ⇒ 6 I n + n I n - 1 = x n e 6 x + c Put n = 10 , 6 I 10 + 10 I 9 = x 10 e 6 x + c

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