MHT CET20244 May 2024Morning ShiftMathematicsIndefinite IntegrationActual
The value of I = x^2 ( a + b x)^2 ~d x is
Options
- A1 b^3 [a+b x+2 a |a+b x|- a^2 a+b x ]+c , (where c is the constant of integration)
- B1 b^3 [a+b x-2 a |a+b x|+ a^2 a+b x ]+c , (where c is the constant of integration)
- C1 b^3 [a+b x-2 a |a+b x|- a^2 a+b x ]+c , (where c is the constant of integration)
- D1 b^3 [a+b x+2 a |a+b x|+ a^2 a+b x ]+c , (where c is the constant of integration)
Correct answer
C. 1 b^3 [a+b x-2 a |a+b x|- a^2 a+b x ]+c , (where c is the constant of integration)
Step-by-step solution
array rl I & = x^2 (a+b x)^2 d x & Let a+b x=t x= t-a b b & d x=d t & = d t b I & = ( t-a b )^2 t^2 d t b & = 1 b^3 (t-a)^2 t^2 d t & = 1 b^3 t^2-2 a t+a^2 t^2 d t & = 1 b^3 [ 1 d t-2 a 1 t +a^2 1 t^2 d t ] array aligned & = 1 b^3 [t-2 a |t|- a^2 t ]+c & = 1 b^3 [a+b x-2 a |a+b x|- a^2 a+b x ]+c aligned