Evaluate the following integral:

We use integration by parts, defining

Then we have,

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Stumbling Robot

A Fraction of a Dot
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Tag: Integration by Parts

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Use integration by parts to prove a given integral formula

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Use integration by parts to evaluate an integral

Evaluate the following integral:

We use integration by parts, defining

Then we have,

Evaluate the following integral:

We use integration by parts, defining

First, we recall that was computed in Example 2 on p. 235 of Apostol which gave

Then we have,

Verify the formula using any method.

for .

*Proof.* As a first step, we write

Then, we integrate by parts with

This gives us

Moving the back to the left we then have,

Let

Compute the following integral (in terms of ):

Starting with the expression for we integrate by parts using

Therefore, we have

Solving for the integral in terms of we have

Let be a function satisfying

Furthermore, let

Compute .

First, using linearity of the integral we write,

Next, we will use integration by parts twice to obtain an expression for the first integral in terms of (and it will turn out the integrals with will cancel which will allow us to solve this problem). For the first integration by parts let,

Therefore,

Now we want to integrate by parts again to evaluate the integral . To that end let

Then we have,

Finally, plugging this back into our original expression we have,

Thus, we have

Evaluate the following integral:

First, we multiply out to get

For the integral on the left we make the substitution , . Therefore,

For the integral on the right we integrate by parts letting

Therefore,

Putting these together we have

- Find a value for such that
- Given that
compute the integral

- To do this we use the expansion/contraction property of the integral (which we proved here). We have
For this equation to be true we must have .

- To compute this we use the formula we established in part (a),
Then, we use integration by parts with

Therefore,

Using integration by parts prove the validity of the formula:

Using this solution evaluate the following integrals

* Proof. * To use integration by parts define the following:

Where the formula for followed by making the substitution , to evaluate

Therefore,

We can then use this solution to evaluate and . For we use the formula with ,

For we use the formula with ,

Using integration by parts, prove the following formula:

Use this solution to evaluate

* Proof. * To apply integration by parts, first define

Where the formula for follows since we can evaluate by making the substitution which implies ; therefore, we have

Then, using integration by parts, we have the following:

Now, applying the formula to , we first note that since this is the case of the above formula with . Therefore,

Then, for we have

Given that

evaluate the integral

To apply integration by parts first define

Then we can use the integration by parts formula to evaluate: