# Integral $\int\frac{(\ln x)^{10}}{x}\,dx$

$$\int\frac{(\ln x)^{10}}{x}\,dx$$ All I know is that I am supposed to substitute $u=\ln x$. But can someone please explain to me how to find the anti derivative of $(\ln x)^{10}$. I think we are supposed to use integration by parts on integrating $\ln x$, but we haven't been taught that yet.

• If you take $u = \ln(x)$, what do you get for $du$? – Mike Pierce Jan 28 '15 at 3:46

## 1 Answer

Hint: Let $u = \ln(x)$, then $\frac{du}{dx} = \frac{1}{x} \implies du = \frac{dx}{x}$

If this is true, what do you see for $$\int \left(\ln(x)\right)^{10}\left(\frac{dx}{x}\right)$$?

• I'm confused.How did du suddenly equal dx/x? – Elsa Jan 28 '15 at 4:28
• I just rearranged some terms. $\frac{\ln(x)^{10}}{x} dx = \ln(x)^{10}\frac{dx}{x}$. I can show more work if you like – jameselmore Jan 28 '15 at 4:31
• oh thank you! let me try it out! – Elsa Jan 28 '15 at 4:32
• Oh I see, this is a common identity, $\frac{d}{dx}\ln(x) = \frac1x$ – jameselmore Jan 28 '15 at 4:33
• wait so the derivative of lnx...OH SHOOT HAHAHA i KEPT DOING THE ANTIDERIVATIVE! Thanks for reminding me hahahaha! now let me try – Elsa Jan 28 '15 at 4:34