There is a trick I’ve seen a few times recently: $$\int_a^b\frac{f(x)}{f(x)+f(a+b-x)}dx=\frac{b-a}2.$$ For example, $$\int_0^{\pi/2}\frac{\sin x}{\sin x+\cos x}=\frac{\pi}4$$ Don’t think it comes up often, except on contests.

@ThomasAndrews I like that, thank you; it's a nice consequence of my 2nd bullet point for definite integrals. You can post that as an answer if you like.

You can integrate $\int_{-\infty}^\infty e^{-x^2}dx$ using a nice trick of integrating the square of the integral with polar coordinates. See en.wikipedia.org/wiki/Gaussian_integral

Also check out "flammable maths" youtube channel, he has a lot of videos featuring weird and wacky integration techniques accessible to strong high school students or first year mathematics degree students.

@MichaelMorrow Thanks for the reference! I've also found 'blackpenredpen's channel really helpful too, he has a crazy range of integrals that he tackles via quite a few different methods.

It is well known, but handy: $\int\frac{f'(x)}{f(x)}\text{d}x=\text{ln}f(x)+C$.

Frullani integral en.wikipedia.org/wiki/Frullani_integral

math.stackexchange.com/questions/2166709/…

The following is My favorite trick for Partial Fractions integration. I explain the idea with some examples: lets say we want to decompose $\frac1{x(x+1)}$ it is equal to $\frac1x-\frac1{x+1}$ . but what about $\frac1{(x-2)(x+7)}$? it is equal to $\frac1{7-(-2)}(\frac1{x-2}-\frac1{x+7})$ (we put the distance between roots in the denominator) . by this method we can decompose lot's of fractions quickly. For example we have $$\int\frac1{x^4-1}dx=\frac12\int\frac1{x^2-1}-\frac{1}{x^2+1}dx=\frac14\ln\left|\frac{x-1}{x+1}\right|-\frac12\tan^{-1}x+C$$

Cont. Sometimes you need to multiply numerator and denominator by something to form these fractions: for example let's consider this example: $$\int\frac1{(x+1)(x^2+4)}dx=\int\frac{x-1}{(x^2-1)(x^2+4)}dx=\frac15\int(x-1)(\frac{1}{x^2-1}-\frac{1}{x^2+4})dx$$Which simplifies to $$\frac15\int\frac1{x+1}-\frac{x-1}{x^2+4}dx$$ And so on

@Svyatoslav Those should be answers, thank you!

I'm glad if it helps. It's a clever idea to find and learn many integration technics which use symmetries and provide shortcut to the solutions. Good luck!

@MichaelMorrow FlammableMaths is a king

All of you seem to be forgetting the most powerful tool: summation! $$1.\int_a^b f(x) dx=\lim_{n\to ∞}\frac{ b-a}{n}\sum_{k=0}^n f(a+k\frac{b-a}{n})$$. 2.Also, there exist many special functions:(functions.wolfram.com) have summation forms and special forms for, say an argument as an integer. 3.Also, the Taylor series for $$f(x)= \sum_{i=0}^ ∞\frac{ f^{(n)=nth derivative}(x) (x-a)^n}{n!}$$ is a great way to find the integral of a function which has a non elementary one.

As you've already written, we have $\displaystyle\int_a^bf(x)dx=\int_a^bf(a+b-x)dx$ from which we can have $$\int_a^bf(x)dx=\frac 12\displaystyle\int_a^b\bigg(f(x)+f(a+b-x)\bigg)dx$$ which sometimes helps. This is such an example.

@FranklinPezzutiDyer Thank you for the recommendation, it looks great!

Yes, IMO Dyer's comment is the best answer, and deserves the bounty if only they would post it as an Answer. There are surely other techniques out there, but for someone whose knowledge essentially ends with standard Calc II methods (but knows them well), Nahin is the best next step.

If your integrand contains a nice-looking nonnegative function, there is a good chance that you can write the integral as $\mathbb{E}[f(X)]$ for some random variable $X$. Sometimes this formulation can be helpful.

Given your knowledge I would recommend you start studying transforms, such as Laplace, Mellin and Fourier transforms; perhaps a little bit about ODEs, because some tricky integrals can be transformed into ODEs after applying Feynman' Trick; also l highly recommend you take a look at Residue Theorem and Friends, a site written by another Stack Exchange member that may simplify your understanding about residual calculus.

@A-LevelStudent I would argue that your question is still a duplicate of the ones you linked to. It would be better for all the integration techniques to be in one place as the other questions don't exclude first-year techniques. If you have trouble understanding the answers there, you should ask another question about what resources you could use to understand them (check out the comment above first!)

@TobyMak I think it's still useful! Say, for someone still in high school to have this post as a reference rather than having to use the other posts which contain a lot of extra information that is way beyond a high schooler's level of understanding. So I feel that it's still beneficial for people to have this question handy!

The best method is feeding the integral to Maple, or Mathematica.