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COMEDK2023Morning ShiftMathematicsIndefinite IntegrationActual

4^x 1-16^x d x is equal to

Options

  1. A( 4) ⁻¹ 4^x+C
  2. B4 4 ⁻¹ 4+C
  3. C1 4 ⁻¹ (4^x )+C
  4. D1 4 ⁻¹ 4^x+C

Correct answer

D. 1 4 ⁻¹ 4^x+C

Step-by-step solution

Let I = 4^x 1-16^x dx . Rewrite the integral as I = 4^x 1-(4^x)^2 dx . Substitute u = 4^x . Then du = 4^x 4 dx , which implies 4^x dx = du 4 . Substituting these into the integral gives I = 1 1-u^2 du 4 . I = 1 4 du 1-u^2 . Using the standard integral du 1-u^2 = ⁻¹ u + C , we get I = 1 4 ⁻¹ u + C . Substituting back u = 4^x , we obtain I = 1 4 ⁻¹ 4^x + C . Answer: 1 4 ⁻¹ 4^x+C

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