# U substitution and integration by parts

I would think because of this

The following problem:

At this stage they should use integration by parts:

However, maybe integration by parts is only useful when one of the parts is e^x ln or a trigonometric formula.

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Integration by parts can be useful whenever the integrand is a product of two functions. But it is not always the easiest method to use. For instance, the integral $\int(u-1)\sqrt{u}du=\int(u^{3/2}-u^{1/2})du$ can easily be solved using the power rule in reverse. Of course, you could solve it using integration by parts as well, but it's just more work than is necessary.

good, thanks.

Using integration by parts might work, but I feel that even if it does work it will be much more work than using u substitution.

Is the problem here that you don't find the u substitution they used to be 'legal'?

Yea, it doesn't seem legal, because I thought you couldn't take the product of two functions in an integrand.

jgens
Gold Member
Yea, it doesn't seem legal, because I thought you couldn't take the product of two functions in an integrand.

You need to work on being more precise. The integrand of $\int_0^1 x^2 \mathrm{d}x$ is the product of two functions but is clearly integrable.