COMEDK2021MathematicsIndefinite Integration
2^ x 1-4^ x d x is equal to
Options
- A( 2) ⁻¹ 2^ x +C
- B1 2 ⁻¹ 2^ x +C
- C1 2 ⁻¹ 2^ x +C
- D2 2 ⁻¹ 2^ x +C
Correct answer
C. 1 2 ⁻¹ 2^ x +C
Step-by-step solution
Let I = 2^ x 1-4^ x dx . Rewrite the integral as I = 2^ x 1-(2^ x )² dx . Substitute u = 2^ x . Then du = 2^ x 2 , dx , which implies 2^ x dx = du 2 . Substituting these into the integral gives I = 1 1-u² du 2 . I = 1 2 du 1-u² . Using the standard integral formula du 1-u² = ⁻¹ u + C , we get I = 1 2 ⁻¹ u + C . Substituting back u = 2^ x , we obtain I = 1 2 ⁻¹ 2^ x + C . Answer: 1 2 ⁻¹ 2^ x +C