TITLE: What other tricks and techniques can I use in integration?
QUESTION [34 upvotes]: So far, I know and can use a reasonable number of 'tricks' or techniques when I solve integrals. Below are the tricks/techniques that I know for indefinite and definite integrals separately.

Indefinite integrals

Standard integrals, such as those of polynomial, trigonometric, logarithmic and exponential functions, including usage of trig identies.
Basic substitution.
Weierstrass and Euler substitutions.
Integration by parts.
$$\int\frac{1}{x+x^n}dx=\int\frac{x^{-n}}{1+x^{1-n}}dx=\frac{1}{1-n}\ln\lvert 1+x^{1-n}\rvert+C$$
$$\int\frac{1}{x^{\frac{a+b}{a+b}}\cdot x^{\frac{a}{a+b}}+x^{\frac{b}{a+b}}}dx=\int \frac{x^{-\frac{b}{a+b}}}{\left(x^{\frac{a}{a+b}}\right)^2+1}dx=\arctan x^{\frac{a}{a+b}}+C$$
Substitution
$u=\frac{1-x}{1+x}$ for integrals involving $\ln$ and/or the bounds $0$ and $1$.
Reduction formulae.
$$\int e^x(f(x)+f'(x))dx=e^xf(x)+C$$
Writing $\sin$'s and $\cos$'s as complex exponentials.
$$\int\frac{a\sin x+b\cos x}{c\sin x+d\cos x}dx=Ax+B\ln\lvert c\sin x+d\cos x\rvert+C$$
where $$A=\frac{ac+bd}{c^2+d^2}~~~B=\frac{bc-ad}{c^2+d^2}$$
which can be found using simultaneous equations.

Definite integrals

Differentiation under the integral sign ('Feynman's technique')
$$\int_a^b f(x)dx=\int_a^bf(a+b-x)dx$$
Usage of power series to evaluate integrals such as $\int_0^1\frac{\ln(1-x)}{x}dx$ and the like.
Making use of even or odd function properties.
(My newest personal favourite) For even functions $f(x)$ and $g(x)$, and an odd function $h(x)$: $$\int_{-a}^a\frac{f(x)}{1\pm g(x)^{h(x)}}dx=\int_{0}^a f(x)~dx$$ which allows us to evaluate wonderful things like
$$\int_{-\infty}^{\infty}\frac{e^{-x^2}}{1+\pi^{\sin x} }dx=\frac{\sqrt{\pi}}{2}$$



Question:
Do you know any other integration techniques or tricks that I can use whose usage don't rely on anything beyond high school calculus* or perhaps the first year of a Mathematics degree course?

I know that a similar question has been asked here and here but I've looked through them and nothing beyond what I have written above was mentioned, apart from some techniques I couldn't understand such as residue calculus and contour integrals.
Many thanks for your help.

*Roughly what I mean by high school level calculus:
INCLUDED

Integration of polynomials and the basic trigonometric functions, such as $\sin x$, $\cos x$, $\tan x$, $\sec x$, $\operatorname{cosec} x$, $\cot x$, $\sec^2 x$, $\sec x\tan x$, $\operatorname{cosec} x\cot x$, $\operatorname{cosec}^2 x.$
Integration of all $x^n$ including $n=1$. Integration of exponentials.
Integration by parts.
Integration using substitution, such as using trigonometric/hyperbolic substitutions, and Weierstrass and Euler substitutions (this also includes integration by 'inspection' which is really just substitution but when the individual doesn't need to substitute anything).
Integration using partial fractions and logarithms, such as $\int\frac{f'(x)}{f(x)}dx$.
Reduction formulae. Ability to understand and use the concepts of even and odd functions in integration. Improper integrals.
Integrating which results in elementary functions.

NOT INCLUDED

Fourier, Laplace and Mellin transforms.
Indefinite integrals that include non-elementary functions in the solution.
Contour integration.
Residue calculus and similar methods.

REPLY [2 votes]: Here are a couple more:

A common trick also is to let the integral in question be $I$, and add it to itself to simplify things.  For example, try computing:

$$I=\int_{0}^{2014}{\frac{\sqrt{2014-x}}{\sqrt{x}+\sqrt{2014-x}}dx}$$
Doing a substitution $u = 2014-x$ and adding the resulting integral to original gives $2I = \int_0^{2014} 1 \,dx$.

Another trick is geometry.  The most common one students encounter is a circle, but here's a different one (from the MIT Integration Bee 2019):

$$\lim_{n \to \infty}\int_{-\infty}^{\infty} e^{-x^{2n}} \,dx.$$If you draw the picture, you just get a rectangle of height $1$ and length $2$, so the answer is $2$.

Riemann sums.  Try computing (without residues)

$$\int_0^{\pi} \ln(1-2a \cos x + a^2) \,dx$$
If you split the interval $[0,\pi]$ into $n$ equally sized intervals, you get a nice factorization that evaluates to $\frac{\pi}{n} \ln \frac{a^{2n}-1}{a^2-1}$, so all you need to do is take the limit.