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# Find the slope and area under the graph for a given function

Let

1. Determine the slope of the graph of at the point with -coordinate 1.
2. Find the volume of the solid of revolution formed by rotating the region between the graph of and the interval about the -axis.

1. To take this derivative, using logarithmic differentiation will be easier,

Then differentiating both sides we have,

So, to find the slope at the point with we evaluate,

2. First, the integral to compute the volume of the solid of revolution is,

To evaluate this we use the partial fraction decomposition,

This gives us the equation

Evaluating at , , and we obtain

Therefore, we have

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

Compute the following integral.

First, we have

Then, we make a substitution, letting

This gives us

Next, we make another substitution, letting and . This gives us,

We can then use partial fractions on the integrand,

This gives us the equation

Solving this for , and we obtain

Therefore,

# Evaluate the integral of (3 – x2)1/2 / x

Compute the following integral.

First, we multiply the numerator and denominator by and do some rearranging to get a friendlier integral,

For the integral on the left we make the substitution , and obtain

Then, for the integral on the right we make the substitution , . This gives us

Therefore for the integral on the right we have

Now, we have to use partial fractions on the integrand,

This gives us the equation

So,

Putting these integrals back into our original expression we have

# Evaluate the integral of 1 / (1+ a cos x) for a > 1

Compute the following integral.

Using the previous exercise we know that with the substitution we obtain

This time, since we make the substitution

This gives us, for ,

Using partial fractions we can rewrite the integrand in this to obtain,

# Evaluate the integral of 1 / (x4 + 1)

Compute the following integral.

First, we need to rewrite the denominator as a product of linear and quadratic terms so that we can use partial fraction decomposition. To that end, we have

Then we write,

This gives us the equation

Here we multiply everything out and equate like powers of . Multiplying out, and grouping like powers of give us,

From this we have the system of equations

Solving this system (sorry, I’m omitting the details of solving the system… there’s nothing tricky in it, but it’s tedious to TeX up into the blog) we obtain

Therefore, we have

Now, we want to simplify the expressions (to get our answer to match the answer Apostol provided in the back of the book). So, we use the identity

Therefore,

Finally, putting this back into the formula we obtained above, we have

# Evaluate the integral of 1 / (x4 – 1)

Compute the following integral.

First, we factor the denominator

To apply partial fraction decomposition we then write,

This gives us the equation

First, we evaluate the equation at and to obtain

Then, using these values of and we evaluate at to obtain

Finally, equating like powers of in the equation we must have . Therefore,

# Evaluate the integral of (1 – x3) / (x(x2 + 1))

Compute the following integral.

First, we factor the numerator and do some rearranging of the integral,

For the remaining integral we use partial fractions, writing

This gives us the equation

First, we evaluate at to obtain

Then we have

Equating like powers of we have and . Therefore,

# Evaluate the integral of 1 / (x4 – 2x3)

Compute the following integral.

First, we want to use partial fraction decomposition so we write

This gives us the equation

Evaluating at and we obtain the values of and ,

Using these values of and and evaluating at and we obtain

Solving this system for and we obtain

Therefore,

# Evaluate the integral of (x4 + 1) / (x (x2 + 1)2)

Compute the following integral.

The denominator is already factored, so we write

This gives us the equation

First, we can evaluate at to obtain

Then, we multiply out and equate like powers of ,

Therefore, equating like powers of , we have the following equations

Solving this system we obtain

Therefore, evaluating the integral we have

# Evaluate the integral of (x+1) / (x3-1)

Compute the following integral.

Since the denominator factors as

we use partial fractions and write

This gives us the equation

First, we evaluate at to find the value of ,

Then, using this value of and evaluating at to obtain the value of ,

Finally, we evaluate at and use the values of and to compute ,

Therefore,