AP EAMCET201921 Apr 2019Evening ShiftMathematicsIndefinite IntegrationActual
∫ d x ( x + 1 ) 2 x 2 + 1 =
Options
- Alog e x + 1 + 1 2 log e x 2 + 1 - 1 x + 1 + C
- Blog e x + 1 - 1 2 log e x 2 + 1 - 1 2 ( x + 1 ) + C
- C1 2 log e x + 1 - 1 4 log e x 2 + 1 + 1 2 ( x + 1 ) + C
- D1 4 log e x + 1 + 1 2 log e x 2 + 1 + 1 x + 1 + C
Correct answer
B. log e x + 1 - 1 2 log e x 2 + 1 - 1 2 ( x + 1 ) + C
Step-by-step solution
The integral expression is given as, I = ∫ d x ( x + 1 ) 2 x 2 + 1 Consider 1 ( x + 1 ) 2 x 2 + 1 = A ( x + 1 ) 2 + B x 2 + 1 + C x + D x 2 + 1 ⇒ 1 = A x 2 + 1 + B ( x + 1 ) x 2 + 1 + ( C x + D ) ( x + 1 ) 2 Substitute x = - 1 , A = 1 2 Substitute x 2 = - 1 . 1 = ( C x + D ) x 2 + 2 x + 1 ⇒ 1 = ( C x + D ) 2 x ⇒ 1 = - 2 C + 2 D x Substitute x = 0 , - 1 2 = C ,   D = 0 Comparing coefficient of x 3 , 0 = B + C ⇒ B = 1 2 Now, I = ∫ 1 2 1 ( x + 1 ) 2 + 1 2 1 x + 1 - 1 2 x x 2 +