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The value of ∫ e x x 1 + e 2 x d x is equal to (where, C is the constant of integration)

Options

Atan - 1 ⁡ 2 e x + C
Bln ⁡ 1 + e x 1 - e x + C
C2 tan - 1 ⁡ e x + C
Dtan - 1 ⁡ x e x + C

Correct answer

C. 2 tan - 1 ⁡ e x + C

Step-by-step solution

Let, I = ∫ e x x 1 + e 2 x d x Substituting e x = t e x 2 x d x = d t I = 2 ∫ d t 1 + t 2 = 2 tan - 1 ⁡ t + C I = 2 tan - 1 ⁡ e x + C

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