How should I do the trigonometric substitution

Integration through trigonometric substitution

Integration through trigonometric substitution is a special case of integration through substitution. This method can always be used when the integrand has a term of the kind , or contains. After applying trigonometric substitution, we get an integral that is easier to integrate than before. In some cases it is only possible to determine the integral through trigonometric substitution.

If one wants to find the area of ​​a circle or an ellipse that corresponds to the integral (a> 0) suffices, one might think that u = a²-x² is the most effective substitution. But the integral is not that easy to solve:


Depending on the term, a different substitution must be selected. There are three ways to substitute trigonometrically:

example 1

Example 2


First we need to choose the appropriate substitution. In this case we can remove the root if we assume x = 3 · tan (θ). This gives us dx = 3 · sec² (θ) dθ.

Now we have to rewrite the integral in terms of the original variable x. Since our substitution is x = 3 · tan (θ), we can solve for tan (θ) and thus get tan (θ) = x / 3. We now only have to resolve sec (θ). Each of the five trigonometric functions can be written in the form of any other. For sec (θ) this would be:

This gives us the end result:

further reading

  1. Bronstein, I. N., & Semendjajew, K. A. (2001). Paperback of Mathematics (5th, revised and expanded edition, unaltered reprint). Thun [and others]: German. ISBN: 978-3817120055.
  2. Glosauer, T. (2015). (High) school mathematics: a stepping stone from high school to university. Wiesbaden [among others]: Springer Spectrum. ISBN: 978-3658058647.