Repeated integration 3 times or more

Question

It is known that if you integrate by parts (twice) say $\int e^{x}cos(x) dx$ you get the original integral again.

Are there any examples of integrals $\int f(x) dx$ where you need to integrate by parts at least 3 times (or more) to get the original integral?

Answer 1

Hint:

If you integrate something, it doesn't matter if it is an integration by parts of any other method, the integrate of a function is independent of the actually used method to calculate it.

Check the complex numbers. Integration by substitution works also for them:

$\int e^{kx} dx=\frac{1}{k}\int e^{kx} dx$

$k$ is a parameter.

If you do this 3 times, you will get $\frac{1}{k^3}\int e^{kx} dx$.

If you want to get back the same, you have to find a $k$, for which $\frac{1}{k^3}=1$. If k is real, then the only possible value is $k=1$. Although in this case it wont't an integration by parts any more.

If we allow complex values, then $k=e^{\frac{2 \pi}{3} i}$ and $k=e^{- \frac{2 \pi}{3} i}$ will be also a solution.

Of course they are complex functions, which may not be allowed by your teacher. But it is already two of them.

You can combine these two complex functions very easily, to get your final, real result.

Answer 2

This happens with $e^x\cos(x)$ because $\cos(x)$ is a solution to $y''+y=0$ (i.e. you have to take its derivative twice to get the negative of the original function). If we want a similar function for which we need to do integration by parts three times, first note that three minus signs come out when doing IBP, and so we want a function such that taking its derivative three times gives us the original function. That is, $$y'''-y=0$$ which has general solution $$y = c_1e^x+c_2e^{\xi x}+c_3e^{\xi^2 x}$$ where $\xi = -\frac{1}{2}+\frac{\sqrt3}{2}i, \xi^2 = -\frac{1}{2}-\frac{\sqrt3}{2}i$. Euler's formula gives us $$y(x) = c_1e^x + e^{-\frac{1}{2}x}\left(A\cos\left(\frac{\sqrt3}{2}x\right) + B\sin\left(\frac{\sqrt3}{2}x\right)\right)$$

Of course, if you actually had to integrate $e^x y(x)$, you would not do integration by parts three times and solve for the integral, although this would be possible. Rather, you would just combine the exponentials, and integrate as normal.