Question

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

Answer 1

`-(30ln(x)+5)/(36x^6)+C`

Explanation:

We have:

`5intln(x)/x^7dx`

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

`intudv=uv-intvdu`

So here, let `u=ln(x)`, so `du=1/xdx`, and `dv=x^-7dx`, and integrate this to see that `v=-1/6x^-6`.

Thus:

`5intln(x)/x^7dx=5[-1/6ln(x)x^-6-int-1/6x^-6(1/x)dx]`

`=-5/6ln(x)/x^6+5/6intx^-7dx`

`=-(5ln(x))/(6x^6)+5/6(-1/6x^-6)+C`

`=-(5ln(x))/(6x^6)-5/(36x^6)+C`

If you want a common denominator:

`=-(30ln(x)+5)/(36x^6)+C`