# Solutions to Problem Set 6

Math 211-03

9-14-2017

[Partial fractions]

1. Compute .

Set . This gives

Set . This gives

Plugging the values for b and c back in yields

Set . This gives

Thus,

Hence,

2. Compute .

Note that

Set up the partial fractions decomposition:

Set . This gives

Set . This gives

Plug and back in to get

From this equation, it follows that .

Alternatively, you can set x to any number besides 0 or 5 in the first equation of the four above, then solve for a (instead of doing the algebra that I did).

Compute the integral:

3. Compute .

Set . This gives

Plugging this back in, I get

Set . This gives

Plugging this back in, I get

Now plugging any number in for x (except -5) allows me to solve for b.

For example, set . Then

With the coefficients found, I can do the integral:

4. Compute .

First, set in the last x-equation. This gives

Save this equation. Now take the original equation and differentiate with respect to x:

I used the Product Rule to differentiate .

Set again. (I'm using because it's easy to do the computations, but you could use other numbers.) This gives

Save this equation. Differentiate the last x-equation:

Set again. This gives

Plug this into to get

Differentiate the last x-equation:

Plug this into to get

Now do the integral:

I integrated the first and third terms using the substitution .

We are what we think. All that we are arises with our thoughts. With our thoughts we make the world. - The Buddha

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