MHT CET202519 Apr 2025Morning ShiftMathematicsIndefinite IntegrationActual
2 x (a+b x)^2 d x=
Options
- A2 a^2 [ (a+ b x)- a a+ b x ]+ c , where c is the constant of integration.
- B-1 a^2 [ (a+ b x)+ a a+ b x ]+ c , where c is the constant of integration.
- C-2 ~b ^2 [ (a+ b x)+ a a+ b x ]+ c where c is the constant of integration.
- D-2 ~b ^2 [ (a+ b x)- a a+ b x ]+ c where c is the constant of integration.
Correct answer
C. -2 ~b ^2 [ (a+ b x)+ a a+ b x ]+ c where c is the constant of integration.
Step-by-step solution
To evaluate the integral 2 x (a+b x)^2 dx , use the substitution u = a + b x . The derivative gives du = -b x dx , so x dx = - 1 b du . With 2x = 2 x x and x = u-a b , the integral becomes: I = 2 u-a b u^2 (- 1 b ) du = - 2 b^2 u-a u^2 du Separate into simpler fractions: I = - 2 b^2 ( 1 u - a u⁻² ) du = - 2 b^2 ( |u| + a u ) + C Substituting back u = a + b x yields: I = - 2 b^2 [ |a + b x| + a a + b x ] + C This matches option C .