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JEE Advanced2026MathematicsDefinite IntegrationActual

The value of the definite integral ₀² 1 3^ x+3 2 , dx is

Options

  1. A1 2
  2. B1 3
  3. C_e 3 3
  4. D_e 3 2

Correct answer

B. 1 3

Step-by-step solution

Let I = ₀² 1 3^x + 3 dx Substitute t = 3^x , which gives dt = 3^x 3 dx dx = dt t 3 . When x = 0 , t = 1 . When x = 2 , t = 9 . The integral becomes: I = ₁⁹ 1 t + 3 dt t 3 = 1 3 ₁⁹ 1 t(t + 3) dt Using partial fractions: 1 t(t + 3) = 1 3 ( 1 t - 1 t + 3 ) I = 1 3 3 ₁⁹ ( 1 t - 1 t + 3 ) dt I = 1 3 3 [ t - (t + 3)]₁⁹ I = 1 3 3 [ ( t t + 3 ) ]₁⁹ I = 1 3 3 ( ( 9 12 ) - ( 1 4 ) ) I = 1 3 3 ( ( 3 4 ) - ( 1 4 ) ) I = 1 3 3 ( 3/4 1/4 ) = 1 3 3 3 I = 1 3

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