Question

Evaluate the integral.
√
S-
5
√x (4+5√x)²
2
dx

Answer 1

Substitute
`y=4+5\sqrt x` and
`dy=\frac5{2\sqrt x}\,dx`. Then the integral is

`\displaystyle \int \frac5{\sqrt x (4+5\sqrt x)^2} \, dx = 2 \int (1)/((4+5\sqrtx)^2) (5)/(2\sqrt x) \, dx = 2 \int y^(-2) \, dy`

By the power rule,

`\displaystyle \int y^(-2) \, dy = -y^(-1) + C`

so that

`\displaystyle \int \frac5{\sqrt x (4+5\sqrt x)^2} \, dx = \boxed{-\frac2{4+5\sqrt x} + C}`