Date modified: Thu 2023-07-06 . 09:49 AM
Date created: Sat 2023-10-07 . 09:02 PM
# Week 2 Tuesday Problems ## Reading. Chapter 7.4. And preview chapter 7.1 ## Problems. Many of these are from chapter 7.4. Please try as many as you can for practice!!!! You can do it ok! 1. For each of the following rational functions, write out the form of the partial fraction decomposition, and determine all the coefficients. 1. $\displaystyle \frac{1}{(x-3)(x+5)}$ 2. $\displaystyle \frac{x+4}{x^2+x-6}$ 3. $\displaystyle \frac{x^2+4}{x^3-3x^2+2x}$ 4. $\displaystyle \frac{x^3+x}{x(3x+2)(x^2+3)^3}$. No need to find coefficient for this one, just the form of pfd. 5. $\displaystyle\frac{x^6}{x^2-4}$. Don't forget to do long division first. 2. Find the following antiderivatives. There are more in the section 7.4 you should try them. Also, try using Desmos to graphically check your answer. 1. $\displaystyle \int \frac{5}{(x-1)(x-4)}dx$ 2. $\displaystyle\int \frac{5x+1}{(2x+1)(x-1)}dx$ 3. $\displaystyle\int \frac{x^2+x+1}{(x+1)^2(x+2)}dx$ 4. $\displaystyle \int \frac{x^3+4x+3}{x^4+5x^2+4}dx$ 5. $\displaystyle\int \frac{5x^4+7x^2+x+2}{x(x^2+1)^2}dx$ 3. Sometimes there are **hidden** rational functions in our integral, by first performing a substitution. Check these out. 1. $\displaystyle \int \frac{dx}{x\sqrt{x-1}}dx$. Try an "obvious" $u$-substitution for a nasty part first, so that at the end it becomes a rational function in $u$. 2. $\displaystyle\int \frac{1}{\sqrt{x}-\sqrt[3]{2}} dx$. Try $u=\sqrt[6]{x}$. (**This was a typo**. Try $u= \sqrt{x}-\sqrt[3]{2}$ instead. ) 3. $\displaystyle \int \frac{1}{1+e^x} dx$. Again try $u$-substitution so that it becomes a rational function integrand. The typo problem is still doable. But it was meant to be this: Find $$ \int \frac{1}{\sqrt{x}-\sqrt[3]{x}}dx $$with $u=\sqrt[6]{x}$. Try both problems if you can.