Integration by the reverse chain rule

The  exercise appears under the Integral calculus Math Mission. This exercise uses u-substitution in a more intensive way to find integrals of functions.

Types of Problems
There is one type of problem in this exercise:


 * 1) Find the indefinite integral: This problem asks for the integral of a function. The student is expected to make an appropriate substitution and simplification, then find the antiderivative using more basic integrals.Ibrcr2.png

Strategies
Knowledge of u-substitution and comfort with algebraic manipulation are encouraged to ensure success on this exercise.
 * 1) The reverse chain rule is really just u-substitution with some minor manipulation of the du.
 * 2) The most common expressions to substitute are the objects by themselves, for example, in the denominator, in the exponent, under the radical, or as an argument to another function.
 * 3) Since the integrals are indefinite in this problem, they can be checked by differentiating the potential answer choices.

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."