Home » Integration by Parts

# Use integration by parts to prove the functional equation of the Gamma function

Recall the definition of the Gamma function:

Using integration by parts, prove the functional equation:

Then use mathematical induction to prove that for positive integers we have

Incomplete.

# Conclude if the given series converges or diverges and justify your conclusion

Test the following series for convergence or divergence. Justify the decision.

First, we make the substitution , which gives us . Then we have

This integral we evaluate using integration by parts with

Therefore, we have

So, for the definite integral from to we have

But then the series is a telescoping series with

Hence,

Hence, the series converges.

# Prove some formulas for integrals of e-t tn

Prove the following integral formulas.

1. .
2. .
3. .
4. Guess and prove a general formula based on parts (a) – (c).

1. Proof. We use integration by parts with

This gives us

2. Proof. We use integration by parts and the result of part (a). Let

This gives us

3. Proof. Again, we use integration by parts, and this time part (b). Let

This gives us

4. Claim:

Proof. The proof is by induction. We have already established the case (and and ). Assume then that the formula holds for some positive integer . We then consider the integral

Integrating by parts, we let

Therefore, integrating by parts and using the induction hypothesis we have,

Therefore, the formula holds for , and hence, for all positive integers

# Determine some properties of the integral of et / t

Let

1. Find all values of such that .
2. Prove that

3. Similar to part (b), find expressions for

1. Recalling the definition of as the integral,

We compute,

Then, since

we have

and

Hence, the set of such that is all real .

2. Proof. We start by computing,

Then, using the translation invariance of the integral,

3. We claim the following formulas hold:

Proof. Making the substitution , we have

Proof. To prove this identity we integrate by parts, letting

Therefore,

Proof. For the final identity, we use the substitution , . This gives us

Now, we use integration by parts with

This gives us

# Explain why ex / x cannot be integrated by parts

Try to use integration by parts to evaluate the integral

If we let

Then we have

If we try to integrate by parts again with and then we’ll end up with another integral, this time with . That will continue, so we’ll just keep getting integrals that we can’t evaluate.

On the other hand, if we try letting

Then we have

Continuing along that route, we’ll keep getting integrals of where keeps getting larger.

In both cases, we keep getting increasing complicated integrals that we can never evaluate.

# Evaluate the integral of (x2 + 5)1/2

Compute the following integral.

First, we want to make the substitution

Therefore, we have

Now, we use integration by parts, letting

Therefore, we have

Moving the back to the left side we have

Plugging this back in above we have,

Then we note that

(If you don’t remember your trigonometry, which I usually don’t, you can figure things like this out by drawing the triangle. Since means you have a right triangle with legs of length and . That means the hypotenuse has length . So, which then gives the above result for .)
Plugging this back in we then have

(Where we absorbed into the constant in the final line, since it is just a constant.)

# Establish the integral formula for (arcsin x)/(x2)

Establish that the following integral formula is correct:

Proof. We can evaluate this integral using integration by parts. Let

Then we have

Next, we must evaluate the integral on the right. For this integral we have

Now, we make a substitution, let , . Therefore,

Combining this with the equation above we then have

# Establish the integral formula for (arcsin x)2

Establish that the following integral formula is correct:

Proof. We’re going to integrate by parts twice to evaluate this integral. First,

So, we have

Now, we evaluate the integral on the right using integration by parts again. Let,

Therefore,

So, putting this back into the equation above we have

# Establish the integral formula for arccsc x

Establish that the following integration formula is correct:

Proof. We integrate by parts, letting

(Where we established the formula for the derivative of in this exercise, Section 6.22, Exercise #5). Therefore,

In the previous exercise we showed that

Therefore,

# Establish the integration formula for arcsec x

Establish that the following integration formula is correct:

Proof. First, we want to integrate by parts. Let

Therefore, we have

Now, to evaluate this integral we break it into pieces (to get rid of the ). Since we have

(Since is only defined for we don’t need to worry about the cases .) We then evaluate the two pieces of the integral separately. If ,

Now, we note that

Therefore, we have an integral of the form and so,

So, for we have

For the case everything is identical except we have a negative sign (since has a negative sign when ) so for we have

Therefore (since if and if ) we have