AP EAMCET201825 Apr 2018Morning ShiftMathematicsIndefinite IntegrationActual
∫ x - 1 ( x + 1 ) x x 2 + x + 1 d x =
Options
- Atan - 1 x 2 + x + 1 x + c
- B2 · tan - 1 x 2 + x + 1 x + c
- Ctan - 1 x 2 + x + 1 x + c
- D2 · tan - 1 x + 1 x + 1 + c
Correct answer
D. 2 · tan - 1 x + 1 x + 1 + c
Step-by-step solution
Let, I = ∫ x - 1 ( x + 1 ) x x 2 + x + 1 d x = ∫ x - 1 x + 1 ( x + 1 ) 2 x 2 x + 1 + 1 x d x = ∫ x 2 - 1 ( x 2 + 1 + 2 x ) x 2 x + 1 + 1 x d x = ∫ x 2 - 1 x x + 1 x + 2 x x + 1 + 1 x d x = ∫ 1 - 1 x 2 x + 1 x + 2 x + 1 x + 1 d x Let x + 1 x + 1 = t 2 ⇒ 1 - 1 x 2 d x = 2 t d t I = ∫ 2 t t 2 + 1 t d t = 2 ∫ 1 t 2 + 1 d t = 2 tan - 1 t + c = 2 tan - 1 x + 1 x + 1 + c