# GRE Subject Test: Math : Substitution

## Example Questions

### Example Question #1 : Substitution

Integrate:

Explanation:

This problem requires U-Substitution.  Let  and find .

Notice that the numerator in  has common factor of 2, 3, or 6.  The numerator can be factored so that the  term can be a substitute. Factor the numerator using 3 as the common factor.

Substitute  and  terms, integrate, and resubstitute the  term.

### Example Question #2 : Substitution

Evaluate the following integral:

Explanation:

To calculate this integral, we could expand that whole binomial, but it would be very time consuming and a bit of a pain. Instead, let's use u substitution:

Given this:

We can say that

Then, plug it back into our original expression

Evaluate this integral to get

Then, replace u with what we substituted it for to get our final answer. Note because this is an indefinite integral, we need a plus c in it.

### Example Question #3 : Substitution

Integrate the following using substitution.

Explanation:

Using substitution, we set  which means its derivative is .

Substituting  for , and  for  we have:

Now we just integrate:

Finally, we remove our substitution  to arrive at an expression with our original variable:

### Example Question #4 : Substitution

Evaluate the following integral:

Explanation:

To calculate this integral, we could expand that whole binomial, but it would be very time consuming and a bit of a pain. Instead, let's use u substitution:

Given this:

We can say that

Then, plug it back into our original expression

Evaluate this integral to get

Then, replace u with what we substituted it for to get our final answer. Note because this is an indefinite integral, we need a plus c in it.