# Rules of integrating by parts?

In my textbook the rule for integrating by parts is

but on a website it is stated that the rule for integrating by parts is

Just wanted to confirm that they are the same rule, right?

Could someone show me how they are the same equation?

Many thanks!

• $v$ in image2 is just $\frac{dv}{dx}$ in image1.
– Yuta
May 9 '19 at 9:30
• thanks for pointing that out, Yuta. Cant believe I didn't notice that... May 9 '19 at 9:36

If you change $$v$$ to $$\frac {dv} {dx}$$ in the second you get the first.

• thanks for replying so soon! I really appreciate it May 9 '19 at 9:35

Both are same because your first image take $$\frac{dv}{dx}$$ in the place of $$v$$ (which is in the second).

$$\int u \frac{dv}{dx} dx = u \int \frac{dv}{dx} dx - \int u'(\int \frac{dv}{dx} dx) dx= uv - \int v \frac{du}{dx} dx$$

• thank you so much for answering so quickly! May 9 '19 at 9:35
• you are welcome @wendy May 9 '19 at 9:39

The by-parts rule comes from the differentiation of a product:

$$(uv)'=u'v+uv'$$ so that

$$\int(uv)'\,dx=uv=\int u'v\,dx+\int uv'\,dx$$ and

$$\int u'v\,dx=uv-\int uv'\,dx.$$

If you denote $$u'$$ as $$w$$, you have $$u=\displaystyle\int w\,dx$$ and

$$\int wv\,dx=\left(\int w\,dx\right)v-\int \left(\int w\,dx\right)v'\,dx.$$

• Thanks for adding to this post, @Yves Daoust I'm very grateful for your contribution :) May 9 '19 at 9:52

They are the same. Just take note however which functions to take as $$u$$ and the other to take as $$dv$$.

According to K.A Straud in Engineering Mathematics, the function you take for $$u$$ should be differentiated to get $$du$$ while the one you take as $$dv$$ should be integrated to have $$v$$. He gave a priority of taking $$u$$ which is thus:

1. $$\ln x$$

2. $$x^n$$

3. $$e^x$$

4. Trigonometric functions.

After, the choice of $$u$$ and $$dv$$, you can then solve using any of the formulae. I hope you understand this?

• I do understand what you're saying. Just wanting to confirm that when you say "Inx" that the number one thing on the list is Natural Log? Thanks for going the extra mile and explaining this to me, I didn't know this before. @Lukgaf May 9 '19 at 9:49
• yes the Natural log. This is because it vanishes immediately. You can find the book here kyumesa.files.wordpress.com/2016/01/… May 9 '19 at 9:51
• Thank you!! This really fleshes out my knowledge, and others who view this post :) May 9 '19 at 9:53
• You are welcome May 9 '19 at 9:55