Method of integration: Trigonometric substitution

carlos84 (72)in Popular STEM • 21 hours ago

The trigonometric substitution method for solving an integral is a method in which we can solve integrals containing radicals of the form:

The objective to be achieved with the application of this method is to be able to solve integrals where the radical can be eliminated from the integrand, for this it is essential to apply the following Pythagorean identities:

Let's take the example where:

Example: solving an integral by the trigonometric substitution method

To solve an integral by applying the trigonometric substitution method, I will place an example of the case we are dealing with in this post, which is the case:

Solve the following integral by the trigonometric substitution method:

If we apply the trigonometric substitution method, the first thing to realize is that it is the case of

Therefore:

As we are already clear about the corresponding equivalences, we then perform the trigonometric substitution:

As could be seen in the previous image, the resulting integral has in the integrand in the numerator part of the fraction dx, however the integral must be in terms of dθ, for this we derive x as follows:

Since x = 3senθ, it implies that:

All that remains is to solve this integral:

Since there is no basic rule of integration that solves this integral, then we apply the following trigonometric identity:

Already the integral of the cosecant squared of theta if solved with the following basic rule of integration:

Therefore:

However, the answer cannot be expressed in terms of trigonometric functions, but we must return the change of variable considering the right triangle:

Since the tangent of theta of the right triangle is equal to opposite cathetus between adjacent cathetus, and the cotangent is the inverse of the tangent, then we substitute in the cotangent of theta the inverse of the tangent, which is adjacent cathetus between opposite cathetus, as follows: