### 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 #31 : Derivatives & 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 #3 : 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 #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 #5 : Integrals

Integrate the following.

**Possible Answers:**

**Correct answer:**

Integration by parts follows the formula:

So, our substitutions will be and

which means and

Plugging our substitutions into the formula gives us:

Since , we have:

, or

### Example Question #6 : Integrals

Evaluate the following integral.

**Possible Answers:**

**Correct answer:**

Integration by parts follows the formula:

In this problem we have so we'll assign our substitutions:

and

which means and

Including our substitutions into the formula gives us:

We can pull out the fraction from the integral in the second part:

Completing the integration gives us:

### Example Question #41 : Derivatives & Integrals

Evaluate the following integral.

**Possible Answers:**

**Correct answer:**

Integration by parts follows the formula:

Our substitutions will be and

which means and .

Plugging our substitutions into the formula gives us:

*Look at the integral: we can pull out the and simplify the remaining as *

.

We now solve the integral: , so:

### Example Question #1 : Integration By Parts

Evaluate the following integral.

**Possible Answers:**

**Correct answer:**

Integration by parts follows the formula:

.

Our substitutions are and

which means and .

Plugging in our substitutions into the formula gives us

*We can pull outside of the integral.*

*Since* , *we have*

### Example Question #1 : Trigonometric Integrals

Integrate the following.

**Possible Answers:**

**Correct answer:**

We can integrate the function by using substitution where so .

Just focus on integrating *sine* now:

The last step is to reinsert the substitution:

### Example Question #1 : Integrals

Integrate the following.

**Possible Answers:**

**Correct answer:**

We can integrate using substitution:

and so

Now we can just focus on integrating *cosine*:

Once the integration is complete, we can reinsert our substitution:

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