Question

`\int\limits {(6)/(√(x) (1+6√(x))^6 ) } \, dx`

Answer 1

Substitute
`y=1+6\sqrt x` and
`dy=(6)/(2\sqrt x)\,dx=\frac3{\sqrt x}\,dx`. Then

`\displaystyle \int \frac6{\sqrt x \left(1 + 6\sqrt x\right)^6} \, dx = \int \frac2{y^6} \, dy \\\\ ~~~~~~~~ = -\frac2{5y^5} + C \\\\ ~~~~~~~~ = \boxed{-\frac2{5\left(1+6\sqrt x\right)^5} + C}`