`int(x^9)/((4x^2+1)^6) dx` is equal to

Options

A. `1/(5x) (4+(1)/(x^(2)))^(-5) + C`
B. `1/(5) (4+(1)/(x^(2)))^(-5) + C`
C. `1/(10x) (1+4x)^(-5) + C`
D. `1/(10)(1/(x^(2))+4)^(-5) + C`

Answer

D

Explanation

Let `I = int (x^(9))/((4x^(2) + 1)) dx = int (x^(9))/(x^(12)(4+1/(x^(2)))^(6))dx`
` = int (dx)/(x^(3)(4+1/x^(2))^(6))`
Put `4 + 1/(x^(2)) = t rArr (-2)/(x^(3)) dx = dt`
`rArr 1/(x^(3)) dx = - 1/2 dt`
`:. I = - (1)/(2) int (dt)/(t^(6)) = - 1/2 [(t^(-6+1))/(-6+1)]+ C`
` = 1/10 [(1)/(t^(5))] + C = 1/10 (4+(1)/(x^(2)))^(-5) + C`