What is the antiderivative of (5 ln(x))/x^(7) ?

Jul 31, 2016

$- \frac{30 \ln \left(x\right) + 5}{36 {x}^{6}} + C$

Explanation:

We have:

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

We will want to use integration by parts, which takes the form:

$\int u \mathrm{dv} = u v - \int v \mathrm{du}$

So here, let $u = \ln \left(x\right)$, so $\mathrm{du} = \frac{1}{x} \mathrm{dx}$, and $\mathrm{dv} = {x}^{-} 7 \mathrm{dx}$, and integrate this to see that $v = - \frac{1}{6} {x}^{-} 6$.

Thus:

$5 \int \ln \frac{x}{x} ^ 7 \mathrm{dx} = 5 \left[- \frac{1}{6} \ln \left(x\right) {x}^{-} 6 - \int - \frac{1}{6} {x}^{-} 6 \left(\frac{1}{x}\right) \mathrm{dx}\right]$

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

$= - \frac{5 \ln \left(x\right)}{6 {x}^{6}} + \frac{5}{6} \left(- \frac{1}{6} {x}^{-} 6\right) + C$

$= - \frac{5 \ln \left(x\right)}{6 {x}^{6}} - \frac{5}{36 {x}^{6}} + C$

If you want a common denominator:

$= - \frac{30 \ln \left(x\right) + 5}{36 {x}^{6}} + C$