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# Prove an integral formula for a rational function in sine and cosine

Given constants such that , prove that

for some constants .

Proof. Define constants and by

(Since we know , so these definitions make sense.) Then

Therefore, we may write,

So, to evaluate the integral we have

For the integral on the right, we make the substitution , so . Therefore,

# 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

# Compute the area and volume of solids of revolution of e-2x

Define the function for all . Let

Compute

1. A(t);
2. V(t);
3. W(t);
4. .

1. The area of the ordinate set on is given by the integral,

2. The volume of the solid of revolution obtained by rotating about the -axis is

3. To compute the volume of the solid of revolution obtained by rotating about the -axis we first find as a function of .

Since , the integral is then from to 1 and we have

4. Finally, using parts (c) and (d) we can compute the limit,

# Find a function given the volume of the solid of revolution it generates

Let be a function continuous on an interval . The volume of the solid of revolution obtained by rotating about the -axis on the interval is given by

for every . Find a formula for the function .

Using the formula for the volume of the solid of revolution generated by a function on an interval we know

Now we differentiate both sides of this equation using the fundamental theorem of calculus on the right-hand side,

# Compute the integral from 0 to x of f(t) for the given functions

Find a formula to compute

for all for the following function .

1. .
2. The function,

3. .
4. the maximum of 1 and .

1. We know from this exercise (Section 5.5, Exercise #13) that

2. If , then over the whole integral, and so

Then, if we have

(Since we have so this equation works. This is the form Apostol wrote these answers as in the back of the book, so I’m getting our answers to match his. I wouldn’t have written them this way otherwise.)

Finally, if we have

Since the formulas for are the same for and are the same we have

for .

3. We consider two cases. If then

If then

4. Since the maximum of 1 and is equal to 1 if and is equal to if we consider three cases (, and ).

For we have

For we have

For we have

# Prove an identity of given finite sums

Prove the identity:

Proof. Using the hint (that ) we start with the expression on the left,

(The interchange of the sum and integral is fine since it is a finite sum. Those planning to take analysis should note that this cannot always be done in the case of infinite sums.) Now, we have a reasonable integral, but we still want to get everything back into the form of the sum on the right so we make the substitution , . This gives us new limits of integration from 1 to 0. Therefore, we have

# Evaluate given integrals in terms of the integral from 0 to 1 of et / (t+1)

Define an integral function

In terms of evaluate the following:

1. .
2. .
3. .
4. .

1. First, we make the substitution so . The bounds of integration are then

Therefore we have,

2. For this one, make the substitution , . The bounds of integration don’t change since and . So we have,

3. To compute this in terms of , we integrate by parts. Let

Therefore we have

4. We use integration by parts again, this time let

Therefore we have,

# Conjecture and prove a statement about a differentiable function satisfying f(x+a) = bf(x)

Let be a function which is differentiable everywhere and which satisfies

for some positive constants and . What can you conclude about such a function ?

(Note: I’m not entirely sure what Apostol wants here since the instruction “what can you conclude” is pretty vague. He does give an “answer” in the back of the book, so I verify that it does have the properties indicated, but I don’t know how you would arrive at that expression just from the question statement. I’ll mark this question as incompletely and hopefully come up with something better in the future.)

Since satisfies the functional equation we can write

which implies

Then computing

Thus, is indeed periodic with period and so

So, this definition of in terms of the periodic function indeed satisfies the functional equation.

# Find continuous functions satisfying given conditions

Find continuous functions which satisfy the given conditions for all .

1. .
2. .
3. .

1. No such function can exist since for we have

2. Taking derivatives of both sides of the given equation we have

3. Again, taking derivatives of both sides we have

at all points such that . (Since is not satisfied by the zero function , we know there are real such that .) Then, integrating

Now, we can solve for by evaluating the given identity at ,

Therefore, we have

# 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