MHT CET202522 Apr 2025Evening ShiftMathematicsIndefinite IntegrationActual
e ^ 2 x ( 2 x 2 x-1) ^2 2 x ~d x=
Options
- Ae ^ 2 x (2 x)+ c , where c is the constant of integration
- B2 e ^ 2 x (2 x)+ c , where c is the constant of integration
- C4 e^ 2 x (2 x)+c , where c is the constant of integration
- D1 2 e ^ 2 x (2 x)+ c , where c is the constant of integration
Correct answer
D. 1 2 e ^ 2 x (2 x)+ c , where c is the constant of integration
Step-by-step solution
To evaluate the integral I = e^ 2x 2x 2x - 1 ^2 2x dx , simplify the integrand first. The numerator separates as 2x 2x ^2 2x - 1 ^2 2x = 2x - ^2 2x , giving I = e^ 2x ( 2x - ^2 2x) dx . Applying integration by parts to e^ 2x 2x dx with u = 2x and dv = e^ 2x dx yields 1 2 e^ 2x 2x + e^ 2x ^2 2x dx . Substituting back, the e^ 2x ^2 2x dx terms cancel, leaving I = 1 2 e^ 2x 2x + C . Alternatively, recognize that for f(x) = 1 2 2x , the integrand is e^ 2x (2f(x) + f'(x)) , so by the standard formula, I = e^ 2x f(x) + C