What is `int (x^(2) + 1)^((5)/(2))x dx` equal to ?
where c is a constant of integration

Options

A. `(x^(2) + 1)^((7)/(2)) + C`
B. `(2)/(7) (x^(2) + 1)^((7)/(2)) + c`
C. `(1)/(7) (x^(2) + 1)^((7)/(2)) + c`
D. None of the above

Answer

C

Explanation

Let `I = int x (x^(2) + 1)^(5//2) dx`
Put `x^(2) + 1 = t`
`2xdx = dt`
`xdx = (dt)/(2)`
`:. I = int (t)^(5//2) (dt)/(2) = (1)/(2) ((t^(7//2))/(7//2)) + c = (1)/(7) (x^(2) + 1)^(7//2) + c`