JEE Main2014MathematicsIndefinite IntegrationActual
If m is a non-zero number and ∫ x 5 m - 1 + 2 x 4 m - 1 x 2 m + x m + 1 3 d x = f x + c , then f x is equal to
Options
- Ax 5 m - x 4 m 2 m x 2 m + x m + 1 2
- B1 2 m x 4 m x 2 m + x m + 1 2
- Cx 5 m 2 m x 2 m + x m + 1 2
- D2 m x 5 m + x 4 m x 2 m + x m + 1 2
Correct answer
B. 1 2 m x 4 m x 2 m + x m + 1 2
Step-by-step solution
∫ x 5 m - 1 + 2 x 4 m - 1 x 2 m + x m + 1 3 d x = f x + c ⇒ ∫ x 5 m - 1 + 2 x 4 m - 1 x 6 m 1 + x - m + x - 2 m 3 d x = f x + c ⇒ ∫ x - m - 1 + 2 x - 2 m - 1 1 + x - m + x - 2 m 3 d x = f x + c . If, 1 + x - m + x - 2 m = t ⇒ - m x - m - 1 + 2 x - 2 m - 1 d x = d t ∴ ∫ - d t m t 3 = f x + c ⇒ - 1 m ∫ t - 3 d t = f x + c Using ∫ x n d x = x n + 1 n + 1 + C ,   n ≠ - 1 ⇒ - 1 m t - 2 - 2 = f x + c ⇒ 1 2 m 1 + x - m + x - 2 m 2 + c