1. ## integrate x^(-2/3)*ln(5x)

Hi all.I have been trying to use integration by parts for this question.

I have got as far as 3x^(1/3)*Ln(5x)-3 (int)x^(1/3)*1/5x.

Im a little stuck on the last bit of integration, i.e, (int)x^(1/3)*1/5x.

I cant see if their is a cancellation to simplify the last bit.

Am i using the correct rule? Can anyone help????

Thanks.

2. ## Re: integrate x^(-2/3)*ln(5x)

The derivative of $\ln{5x}$ is not $\frac{1}{5x}$.

Either way, surely you know what $\frac{x^n}{x}$ is?

3. ## Re: integrate x^(-2/3)*ln(5x)

Let y = ln u
u = 5x
du/dx = 5
dy/du = 1 / u
dy/dx = (dy/du) X (du/dx)
dy/dx = (1/u) X 5
dy/dx = (1 / 5x) X 5
dy/dx = 1 / x
Im failing to see where to use this.

x^n/x = x^n-1

4. ## Re: integrate x^(-2/3)*ln(5x)

Originally Posted by bignaughtydog
Hi all.I have been trying to use integration by parts for this question.

I have got as far as 3x^(1/3)*Ln(5x)-3 (int)x^(1/3)*1/5x.

Im a little stuck on the last bit of integration, i.e, (int)x^(1/3)*1/5x.

I cant see if their is a cancellation to simplify the last bit.

Am i using the correct rule? Can anyone help????

Thanks.
Using integration by parts: \displaystyle \begin{align*} \int{u\,dv} = u\,v - \int{v\,du} \end{align*} with \displaystyle \begin{align*} u = \ln{(5x)} \implies du = \frac{1}{x}\,dx \end{align*} and \displaystyle \begin{align*} dv = x^{-\frac{2}{3}}\,dx \implies v = 3x^{\frac{1}{3}} \end{align*} we have

\displaystyle \begin{align*} \int{x^{-\frac{2}{3}}\ln{(5x)}\,dx} &= 3x^{\frac{1}{3}}\ln{(5x)} - \int{3x^{\frac{1}{3}}\cdot \frac{1}{x}\,dx} \\ &= 3x^{\frac{1}{3}}\ln{(5x)} - \int{3x^{-\frac{2}{3}}\,dx} \\ &= 3x^{\frac{1}{3}}\ln{(5x)} - 9x^{\frac{1}{3}} + C \end{align*}

5. ## Re: integrate x^(-2/3)*ln(5x)

Originally Posted by bignaughtydog
Hi all.I have been trying to use integration by parts for this question.

I have got as far as 3x^(1/3)*Ln(5x)-3 (int)x^(1/3)*1/5x.

Im a little stuck on the last bit of integration, i.e, (int)x^(1/3)*1/5x.

I cant see if their is a cancellation to simplify the last bit.

Am i using the correct rule? Can anyone help????

Thanks.
WA , click "show steps"

6. ## Re: integrate x^(-2/3)*ln(5x)

Thank you thank you priceps and prove it.That was the answer i got after finding the derivative of ln(5x).Ridley did give me a nudge in that direction also.Thanks for that Ridley also.....