Integration by Substitution with a Definite Integral

In this topic:

(1) Proposition (Integration by Substitution) If is a continuous function of and is a differentiable function of , then

(2) Example (Integration by Substitution with a Definite Integral) Evaluate the definite integral

(3) Example (Integration by Substitution with a Definite Integral) Evaluate the definite integral

(4) Example (Integration by Substitution with a Definite Integral) Evaluate the definite integral

(5) Example (Integration by Substitution with a Definite Integral) Evaluate the definite integral

(6) Example (Integration by Substitution with a Definite Integral) The slope at each point on the graph of is given by What is if the graph passes through the point

(7) Example (Integration by Substitution with a Definite Integral) Evaluate

Proposition (Integration by Substitution) If is a continuous function of and is a differentiable function of , then

Example (Integration by Substitution with a Definite Integral) Evaluate the definite integral

Solution. Let and so Then

Example (Integration by Substitution with a Definite Integral) Evaluate the definite integral

Solution. Let then and so

Example (Integration by Substitution with a Definite Integral) Evaluate the definite integral

Solution. Using the fundamental theorem of calculus,

For the seond integral we let and so Therefore,

Example (Integration by Substitution with a Definite Integral) Evaluate the definite integral

    Solution. First we rewrite,


Now let and then and so

Example (Integration by Substitution with a Definite Integral) The slope at each point on the graph of is given by What is if the graph passes through the point

Solution. We are given and so Let then

Now allows us to determine by way of and therefore

Example (Integration by Substitution with a Definite Integral) Evaluate

Solution. First factor by grouping to obtain,

Then let and so and then

Then let and so and then