# Are there functions that cannot be integrated using integration by parts?

Nov 5, 2015

Yes, there are infinitely many functions that cannot be integrated with a close form integral.

#### Explanation:

If you can integrate $\int f \left(x\right) \mathrm{dx}$ at all, then you can, in a trivial sense, integrate by parts.

$u = 1$ and $\mathrm{dv} = f \left(x\right) \mathrm{dx}$ so $\mathrm{du} = 0 \mathrm{dx}$ and $v = \int f \left(x\right) \mathrm{dx}$

$u v - \int v \mathrm{du} = 1 \int f \left(x\right) \mathrm{dx} - \int \left[\int f \left(x\right) \mathrm{dx}\right] 0 \mathrm{dx}$

Examples of integrals without closed form include

$\int {e}^{{x}^{2}} \mathrm{dx}$, $\int \cos \frac{x}{x} \mathrm{dx}$ and $\int \sin \frac{x}{x} \mathrm{dx}$