Integration using trigonometric substitution

The  exercise appears under the Integral calculus Math Mission. This exercise practices trigonometric substitutions to help simply integrals.

Types of Problems
There are two types of problems in this exercise:


 * 1) Determine the correct trig substitution to use: This problem provides an integral and a list of possible substitutions. The student is asked to determine which substitution would be most correct to simplify the integral.Iuts1.png
 * 2) Describe things equivalent to the trig substitution: This problem performs a particular substitution and makes the appropriate reference triangle. The student is expected to use the reference triangle to simplify the integral or describe other relationships that are true based on that substitution.Iuts2.png

Strategies
Knowledge of reference triangles and trigonometric identities are encouraged to ensure success on this exercise.
 * 1) The Pythagorean trigonometric identities can be used to help recognize the appropriate substitutions.
 * 2) Once a substitution is made, a reference triangle can be sketched to help increase accuracy on these problems.
 * 3) For sqrt(1-x^2) a sine or cosine should be substituted.
 * 4) For 1+x^2, a tangent function should be substituted.

Real-life Applications

 * 1) Integral techniques are important for finding any integral that is not immediately set up as one that is recognizable.
 * 2) Integration has many applications in the sciences and economics to find "totals."