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JEE Main202127 Aug 2021Evening ShiftMathematicsIndefinite IntegrationActual

∫ 2 e x + 3 e - x 4 e x + 7 e - x d x = 1 14 u x + v log e 4 e x + 7 e - x + C , where C is a constant of integration, then u + v is equal to

Correct answer

0

Step-by-step solution

I = ∫ 2 e x + 3 e - x 4 e x + 7 e - x d x = ∫ 2 e x + 3 4 e 2 x + 7 d x Here 2 e 2 x + 3 = A 8 e 2 x + B 4 e 2 x + 7 ⇒ 2 e 2 x + 3 = ( 8 A + 4 B ) e 2 x + 7 B ⇒ B = 3 7   &   A = 1 28 I = ∫ 1 28 8 e 2 x + 3 7 4 e 2 x + 7 4 e 2 x + 7 d x I = 1 28 ln 4 e 2 x + 7 + 3 7 x + C I = 1 28 ln ∣ e x 4 e x + 7 e - x ∣ + 3 7 x + C ⇒ u = 13 2 & v = 1 2 ⇒ u + v = 13 2 + 1 2 = 7

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