There's always Feynman's trick. It's more common amongst physicists; Feynman, afterall was a physicist. Although quite standard now, back in Feynman's hayday he was able to solve many complicated integrals by adding another variable, using an auxiliary function, solving a basic integral involving the auxiliary function, and then differentiating said variable to get the correct integral. This is common now, but when Feynman would use it, people wouldn't understand how he'd solved such complicated integrals so fast. It's very diverse too, in that it applies in many scenarios. One example I always found interesting, off the top of my head, was using it to prove that the Gamma function interpolates the factorial in one line:

$$\int_0^\infty e^{-x}x^{n}\,dx = (-1)^{n}\frac{d^{n}}{dt^{n}}\Big{|}_{t=1}\int_0^\infty e^{-tx}\,dx = \frac{d^{n}}{dt^{n}}\Big{|}_{t=1} \frac{(-1)^{n}}{t} = n!$$

though is of no way limited to this single, obvious, instance. One can solve the $\text{sinc}$ integral using this. One can simplify the amount of work one needs by using this little trick.

It is a trick in the sense that it's very simple. It's a little magic, where persons usually go "where'd you think of using that extra variable and that second function and taking the derivative?" It's basic and easy to understand, isn't a theorem or a lemma really, it's just a technique one tries when solving integrals. And, as a kicker, it can go very far with little work; Feynman made a living off using this trick.

There's always Feynman's trick. It's more common amongst physicists; Feynman, afterall was a physicist. Although quite standard now, back in Feynman's hayday he was able to solve many complicated integrals by adding another variable, using an auxiliary function, solving a basic integral involving the auxiliary function, and then differentiating said variable to get the correct integral. This is common now, but when Feynman would use it, people wouldn't understand how he'd solved such complicated integrals so fast. It's very diverse too, in that it applies in many scenarios. One example I always found interesting, off the top of my head, was using it to prove that the Gamma function interpolates the factorial in one line:

$$\int_0^\infty e^{-x}x^{n}\,dx = (-1)^{n}\frac{d^{n}}{dt^{n}}\Big{|}_{t=1}\int_0^\infty e^{-tx}\,dx = \frac{d^{n}}{dt^{n}}\Big{|}_{t=1} \frac{(-1)^{n}}{t} = n!$$

though is of no way limited to this single, obvious, instance. One can solve the $\text{sinc}$ integral using this. One can simplify the amount of work one needs by using this little trick.

It is a trick in the sense that it's very simple. It's a little magic, where persons usually go "where'd you think of using that extra variable and that second function and taking the derivative?" It's basic and easy to understand, isn't a theorem or a lemma really (putting down hypotheses or conditions blurs its simplicity). It's just a technique one tries when solving integrals; like partial fractions or trigonometric substitution; except it takes a while to get a hang of using, and at first seems extraneous. I think that's an important fact to something being a trick: it isn't obvious, but it's easy, and it takes a while to get in the hang of using it. As a kicker, Feynman's trick can go very far with little work; Feynman made a living off using this trick.