W03 Regular

Due date: Sunday 2/1, 11:59pm

Partial fractions

01

04

Partial fractions - irreducible quadratic

Compute the integral:

$$\int \frac{x^2}{x^2+9}dx$$

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Solution

Solutions---0060-04

02 $★$

05

Partial fractions - long division

Compute the integral:

$$\int \frac{x^2-x+1}{x^2+x}dx$$

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Solution

Solutions---0060-05

03

06

Partial fractions - big generic

Give the generic partial fraction decomposition (no need to solve for the constants):

$$\frac{x+2}{(x^2+2){(x-1)}^3(x^2-9)}=\frac{A}{?}+?$$

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Solution

Solutions---0060-06

04 $★$

07

Partial fractions - linear and quadratic

Compute the integral:

$$\int \frac{5x^2-5x+14}{(x-2)(x^2+4)}dx$$

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Solution

Solutions---0060-07

05 $★$

08

Partial fractions - repeated factor

Compute the integral:

$$\int \frac{1}{x{(x-1)}^3}dx$$

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Solution

Solutions---0060-08

06

09

Partial fractions - rationalize first

For each of these integrals, make a $u$-substitution that changes the integrand into a rational function. Write the integral in terms of $u$ for your answer. You do not have to compute the $u$-integral.

(a) $\int \frac{1}{e^x-1}dx$

(b) $\int \frac{\sqrt{x}}{x-1}dx$

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Solution

Solutions---0060-09

Simpson’s Rule

07

02

Simpson’s Rule for volume by shells

Use Simpson’s Rule with $n=6$ to compute the volume of the solid obtained by revolving the pictured region about the $y$-axis. Can you do it without using a calculator?

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Solution

Solutions---0070-02

08 $★$

03

Area of a garden bed

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Solution

Solutions---0070-03