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The integral I = ∫ x e x 2 sin ⁡ x 2 + cos ⁡ x 2 d x = f x + c , (where, c is the constant of integration). Then, f x can be

Options

  1. Ae x sin ⁡ x 2
  2. Be x 2 sin ⁡ x
  3. Ce x 2 sin ⁡ x 2 2
  4. D1 2 e x 2 sin ⁡ x 2

Correct answer

D. 1 2 e x 2 sin ⁡ x 2

Step-by-step solution

Let, x 2 = t ⇒ 2 x d x = d t ∴ I = 1 2 ∫ e t sin ⁡ t + cos ⁡ t d t = 1 2 e t ⋅ sin ⁡ t + c = 1 2 e x 2 sin ⁡ x 2 + c As, ∫ e x f x + f ′ x d x = e x ⋅ f x + c

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