### All GRE Subject Test: Math Resources

## Example Questions

### Example Question #1 : Integrals

Integrate:

**Possible Answers:**

**Correct answer:**

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 #1 : Integrals

Evaluate the following integral:

**Possible Answers:**

**Correct answer:**

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 #1 : Integrals

Integrate the following using substitution.

**Possible Answers:**

**Correct answer:**

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:

**Possible Answers:**

**Correct answer:**

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.