# Are there functions for which the cyclic integration-by-parts technique does not work?

There are a lot of functions where you can use what my teacher has described as the 'cyclic' method of integration. An example is $$\int e^x\sin x\,dx$$ where you designate $u=\sin x$ and $dv=e^x\,dx$. You do integration by parts and arrive at $$\int e^x \sin x \,dx=e^x \sin x - \int e^x \cos x\,dx$$ Now do it again, and you eventually arrive at $$\int e^x \sin x\, dx=g(x)-n\int e^x \sin x \,dx$$ where $g(x)$ is a function of $x$. Re-arrange this and solve.

This also works for, say $\int e^x \cos x \,dx$.

Are there any related integrals involving $\sin x$ or $\cos x$ where this technique doesn't work? (Excluding those such as $\int x^n \sin x\, dx$, where there isn't a problem.)

• This is typically only a property of periodic functions whose derivatives behave cyclically in some sense. There are a lot of functions for which it does not work. But for functions like sin and cos and e this is a nice property – Eoin Oct 25 '14 at 22:19
• Try $\int \cos x \cos x \, dx$. – Hans Lundmark Oct 26 '14 at 5:29

Basically this only works for products of functions whose derivatives repeat:

$$\frac{d^n}{dx^n} f(x) = f(x)$$

for some $n$

$\sin x$, $\cos x$ and $e^x$ have this property, so only products of these functions will work. Otherwise, it will not work. Examples:

$$\int \ln x \, \cos x \, dx$$

$$\int e^{\sin x} \, dx$$

$$\int \frac{\sin x}{x} dx$$

• Awesome. This is just what I needed ("Just what I need-ed!") – HDE 226868 Oct 31 '14 at 1:33
• I don't know why you need to find integrals that can't be solved though. Wouldn't it be more intuitive to know in which cases it does work? – Dylan Oct 31 '14 at 1:35
• True, but I found I could more easily figure out in which cases it does work than which cases it doesn't work. – HDE 226868 Oct 31 '14 at 1:36