[Math] What other tricks and techniques can I use in integration

big-listcalculusindefinite-integralsintegrationsoft-question

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.

Best Answer

As a high school student, most of the tricks I'm aware of were already stated by you, or in the comments. However, there's one more trick that I don't think anyone has mentioned: Integrating an inverse function.

$$\int\!f^{-1}(x)\ dx = x\cdot\!f^{-1}(x)\ - F(f^{-1}(x))\ + c$$where $$F(x) = \int\!f(x)\ dx$$

So for instance, if you wish to find $\int\cos^{-1}(x)\ dx,$ you will have $$f(x)= \cos x$$ and $$F(x) = \int\cos x\ dx = \sin x\ (+c)$$

So to find $\int\cos^{-1}(x)\ dx,$ use the formula as the follows:

$$\int\cos^{-1}(x)\ dx = x\cdot\cos^{-1}(x)\ - \sin(\cos^{-1}(x))\ + c$$ $$= x\cdot\cos^{-1}(x)\ - \sqrt {1-x^2}\ + C$$

I personally like this trick as it can be generalized to any inverse function. A simple way to prove it would be using the Chain Rule but it's a really nice formula that avoids working things out from scratch every time.

Hope that helped to add to your list :)

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