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COMEDK2022MathematicsIndefinite Integration

3^x 1-9^x d x is equal to

Options

  1. A( 3) ⁻¹ 3^x+C
  2. B1 3 ⁻¹ (3^x )+C
  3. C1 3 ⁻¹ 3^x+C
  4. D3 3 ⁻¹ 3^x+C

Correct answer

C. 1 3 ⁻¹ 3^x+C

Step-by-step solution

Let I = 3^x 1-9^x dx . Rewrite the denominator as 1-(3^x)^2 . Let u = 3^x . Then du = 3^x 3 dx , which implies 3^x dx = du 3 . Substituting these into the integral: I = 1 1-u^2 du 3 I = 1 3 du 1-u^2 Using the standard integral formula dx 1-x^2 = ⁻¹ x + C , we get: I = 1 3 ⁻¹ u + C Substituting u = 3^x back: I = 1 3 ⁻¹ 3^x + C . Answer: 1 3 ⁻¹ 3^x+C

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