# GRE Subject Test: Math : Integrals

## Example Questions

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### Example Question #1 : Integrals

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 #31 : Derivatives & Integrals

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 : Integrals

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 #1 : Integrals

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 #5 : Integrals

Integrate the following.

Explanation:

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.

Explanation:

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.

Explanation:

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.

Explanation:

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.

Explanation:

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.

Explanation:

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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