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JEE Main202131 Aug 2021Morning ShiftMathematicsIndefinite IntegrationActual

The integral ∫ 1 ( x - 1 ) 3 ( x + 2 ) 5 4 d x is equal to : (where C is a constant of integration)

Options

  1. A3 4 x + 2 x - 1 5 4 + C
  2. B4 3 x - 1 x + 2 1 4 + C
  3. C4 3 x - 1 x + 2 5 4 + C
  4. D3 4 x + 2 x - 1 1 4 + C

Correct answer

B. 4 3 x - 1 x + 2 1 4 + C

Step-by-step solution

I = ∫ d x ( x - 1 ) 3 ( x + 2 ) 5 4 I = ∫ d x ( x - 1 ) 3 4 ( x + 2 ) 5 4 I = ∫ d x ( x - 1 ) 2 x + 2 x - 1 5 4 Let x + 2 x - 1   = t Differentiate on both sides x - 1 - x + 2 x - 1 2 . d x   =   d t ⇒ d x x - 1 2   =   - d t 3 I = - 1 3 ∫ d t t 5 4 =   - 1 3 t - 5 4 + 1 1 - 5 4 + C = 4 3 t - 1 4 + C = 4 3 1 t 1 4 + C =   4 3 x - 1 x + 2 1 4 + C

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