Integration by Parts - GRE Quantitative Reasoning
Card 0 of 16
Evaluate the following integral.

Evaluate the following integral.
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:


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:
Compare your answer with the correct one above
Integrate the following.

Integrate the following.
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

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
Compare your answer with the correct one above
Evaluate the following integral.

Evaluate the following integral.
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:


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:
Compare your answer with the correct one above
Evaluate the following integral.

Evaluate the following integral.
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


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
Compare your answer with the correct one above
Evaluate the following integral.

Evaluate the following integral.
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:


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:
Compare your answer with the correct one above
Integrate the following.

Integrate the following.
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

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
Compare your answer with the correct one above
Evaluate the following integral.

Evaluate the following integral.
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:


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:
Compare your answer with the correct one above
Evaluate the following integral.

Evaluate the following integral.
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


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
Compare your answer with the correct one above
Evaluate the following integral.

Evaluate the following integral.
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:


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:
Compare your answer with the correct one above
Integrate the following.

Integrate the following.
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

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
Compare your answer with the correct one above
Evaluate the following integral.

Evaluate the following integral.
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:


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:
Compare your answer with the correct one above
Evaluate the following integral.

Evaluate the following integral.
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


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
Compare your answer with the correct one above
Evaluate the following integral.

Evaluate the following integral.
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:


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:
Compare your answer with the correct one above
Integrate the following.

Integrate the following.
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

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
Compare your answer with the correct one above
Evaluate the following integral.

Evaluate the following integral.
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:


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:
Compare your answer with the correct one above
Evaluate the following integral.

Evaluate the following integral.
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


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
Compare your answer with the correct one above