# How do you integrate (5 ln(x))/x^(7)?

May 13, 2018

The answer is $= - \frac{5}{6} \ln \frac{x}{x} ^ 6 - \frac{5}{36 {x}^{6}} + C$

#### Explanation:

Apply the Integration by parts

$\int u v ' = u v - u ' v$

The integral is

$I = \int \frac{5 \ln x \mathrm{dx}}{x} ^ 7 = 5 \int \frac{\ln x}{x} ^ 7$

$u = \ln x$, $\implies$, $u ' = \frac{1}{x}$

$v ' = {x}^{-} 7$, $\implies$, $v = - \frac{1}{6 {x}^{6}}$

Therefore,

$I = - \frac{5}{6} \ln \frac{x}{x} ^ 6 + \frac{5}{6} \int \frac{\mathrm{dx}}{x} ^ 7$

$= - \frac{5}{6} \ln \frac{x}{x} ^ 6 - \frac{5}{36 {x}^{6}} + C$