AP EAMCET202317 May 2023Evening ShiftMathematicsIndefinite IntegrationActual
d x (2 a x+x^2 )^ 3 2 =
Options
- A1 a^2 ( x+a 2 a x+x^2 )+C
- B1 a^2 ( x-a 2 a x+x^2 )+C
- C-1 a^2 ( x-a 2 a x+x^2 )+C
- D-1 a^2 ( x+a 2 a x+x^2 )+C
Correct answer
D. -1 a^2 ( x+a 2 a x+x^2 )+C
Step-by-step solution
Let I = d x (2 a x+x^2 )^ 3 / 2 = d x ((x+a)^2-a^2 )^ 3 / 2 Let t=x+a d t=d x I = d t (t^2-a^2 )^ 3 / 7 Put t=a p aligned & d t d p =a p p d t=a p p d p & I = a p p (a^2 2 p-a^2 )^ 3 / 2 d p & = 1 a^2 p p ( ^2 p )^ 3 / 2 d p= 1 a^2 p ^2 p d p & =- 1 a^2 ( 1 p )+c=- 1 a^2 ( 1 ( ⁻¹ t a ) )+c & =- 1 a^2 ( 1 ( (a) t^2 a^2 -1 t )+c=- 1 a^3 ( t t^2 a^2 -1 )+c . & =- 1 a^3 ( x+a ( x + a )^2 a^2 -1 )+c=- 1 a^2 ( x+a x^2+2 a x )+c aligned