Trigonometric Substitution: Sin Substitution Example 1| (Ali BA)

You may wonder how did they construct the table of the integration. One of the answers is that they use trig substitution.

Trig substitution has two types:
1) sin substitution x=a*sin(theta). In case of the integrand has the form: sqrt(a^2-x^2)
To simplify the integrand, we use the trig identity: cos^2+sin^2=1
2)tan substitution x=a*tan(theta). In case of the integrand has the from: a^2+x^2 or square root of that.
To simplify the integrand, we use tan=sin/cos and then we use the trig identity: cos^2+sin^2=1

Note: in some cases, we may need to complete the square of the dominator before we use the trig substitution.

Steps to solve integrals using trig substitution:
1- You make sure that the integrand is in the form of sqrt(a^2-x^2), a^2+x^2 or square root of that.
2- You make the substitution. What is x and dx.
sin substitution x=a*sin(theta). In case of the integrand has the form sqrt(a^2-x^2)
tan substitution x=a*tan(theta). In case of the integrand has the from a^2+x^2 or square root of that
3- You take the integrand and use the trig substitution to re-write it. Then, simplify.
4- You take the integral and use the trig substitution to re-write it. Then, simplify.
5- After you find the antiderivative of step 4, you back substitute the function of theta from step 2.

Remember:
1)arctan(x)=tan^-1(x), arcsin(x)=sin^-1(x)
2)arctan(tan(x))=x, arcsin(sin(x))=x

Table of integration:

If you have any questions or suggestions, let me know in the comments below.

Material in this video is taken from Calculus: Single Variable, 6th Edition by (Hallet, McCallum, Gleason, et al.)