High School Math : Finding Integrals by Substitution

Example Questions

Example Question #1 : Finding Integrals By Substitution

Determine the indefinite integral:

Explanation:

, so this can be rewritten as

Set . Then

and

Substitute:

The outer factor can be absorbed into the constant, and we can substitute back:

Example Question #1 : Finding Integrals By Substitution

Evaluate:

Explanation:

Set . Then

and

Also, since , the limits of integration change to  and .

Substitute:

Example Question #41 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Determine the indefinite integral:

Explanation:

Set . Then

.

and

The integral becomes:

Substitute back:

Example Question #1 : Finding Integrals By Substitution

Explanation:

This integral will require a u-substitution.

Let .

Then, differentiating both sides, .

We need to solve for dx in order to replace all x terms with u terms.

.

This is a little tricky because we stilll have x and u terms mixed together. We need to go back to our original substitution.

Now we have an integral that looks more manageable. First, however, we can't forget about the bounds of the definite integral. We were asked to evaluate the integral from  to . Because , the bounds will change to  and .

Essentially, we have made the following transformation:

.

The latter integral is easier to evaluate.

At this point, we can separate the integral into two smaller integrals.

.

The integral evaluates to -2, so now we just need to worry about the other integral. This will require the use of partial fraction decomposition. We need to rewrite  as the sum of two fractions.

We need to solve for the values of A and B.

This means that  and . This is a relatively simple system of equations to solve, so I won't go into detail. The end result is that .

Let's now go back to the integral .

Distribute the 2 to both integrals and separate it into two integrals.

.

Remember we need to add this value back to the value of , which we already determined to be -2.

Example Question #1 : Finding Integrals By Substitution

Evaluate:

Explanation:

Set .

Then and .

Also, since , the limits of integration change to  and .

Substitute: