# Thread: Indefinite Integration Using Substitution

1. ## Indefinite Integration Using Substitution

Find $\displaystyle \int \frac{(lnx)^7}{x} dx$.

Let $\displaystyle u = lnx$

Let $\displaystyle du = \frac{1}{x} dx => dx = x du$

$\displaystyle = \int \frac{1}{x} (lnx)^7 dx$

$\displaystyle = \int (\frac{1}{x}) ((lnx)^7) (x) (du)$

$\displaystyle = \frac{u^8}{8} + c$

... I stopped right here because I'm not getting the right answer... which is...

$\displaystyle = \frac{1}{4}u^4 + c$

I don't get where $\displaystyle \frac{1}{4}$ and $\displaystyle u^4$ comes from??? I thought I'm following the antiderivative rules???

2. Originally Posted by Macleef
Find $\displaystyle \int \frac{(lnx)^7}{x} dx$.

Let $\displaystyle u = lnx$

Let $\displaystyle du = \frac{1}{x} dx => dx = x du$

$\displaystyle = \int \frac{1}{x} (lnx)^7 dx$

$\displaystyle = \int (\frac{1}{x}) ((lnx)^7) (x) (du)$

$\displaystyle = \frac{u^8}{8} + c$

... I stopped right here because I'm not getting the right answer... which is...

$\displaystyle = \frac{1}{4}u^4 + c$

I don't get where $\displaystyle \frac{1}{4}$ and $\displaystyle u^4$ comes from??? I thought I'm following the antiderivative rules???
0.25u^4+C is wrong, your solution is absolutely correct.

3. don't forget to back-substitute, the solution is $\displaystyle \frac 18 ( \ln x)^8 + C$