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# Week 2 Wednesday Problems
## Reading.
Chapter 7.1. And preview chapter 7.2, 7.3
You got this ! ! !
## Problems.
1. Evaluate the following using integration by parts. More practice in text 7.1
1. $\displaystyle \int x e^{2x}dx$
2. $\displaystyle \int \sqrt{x} \ln(x) dx$
3. $\displaystyle\int x \cos(4x)dx$
4. $\displaystyle\int \arcsin(x) dx$
5. $\displaystyle\int x\ln(x)dx$
6. $\displaystyle\int \frac{\ln(x)}{x^2}dx$
7. $\displaystyle\int e^{2x}\cos(3x)dx$
8. $\displaystyle\int \arccos(x)dx$
9. $\displaystyle\int \ln(\sqrt{x})dx$
10. $\displaystyle\int x \cosh(x)dx$
11. $\displaystyle\int\arctan(\frac{1}{x})dx$
12. $\displaystyle\int\cos(x)\sinh(x)dx$
2. Sometimes you want to do a $u$-substitution first, and then apply integration by parts.
1. $\displaystyle\int e^{\sqrt{x}}dx$
2. $\displaystyle\int x^3\cos(x^2)dx$
3. $\displaystyle\int \cos(\ln(x))dx$
4. $\displaystyle\int x\ln(1+x)dx$
5. $\displaystyle\int \frac{\arcsin(\ln(x))}{x}dx$
3. Use the reduction formulas derived from class / notes to compute the following (so that there is no integral symbol at the end):
1. $\displaystyle\int\sin^4(x)dx$
2. $\displaystyle\int\sin^3(x)dx$
3. $\displaystyle\int \frac{1}{(x^2+1)^3}dx$
4. Use integration by parts to establish the following reduction formulas:
1. $\displaystyle\int\cos^n(x)dx=\frac{1}{n}\cos^{n-1}(x)\sin(x)+ \frac{n-1}{n}\int\cos^{n-2}(x)dx$
2. $\displaystyle\int(\ln x)^ndx=n(\ln x)^n - n \int (\ln x)^{n-1}dx$
3. $\displaystyle\int x^n e^x dx = x^n e^x -n\int x^{n-1}e^x dx$
4. $\displaystyle \int \tan^n(x)dx= \frac{\tan^{n-1}(x)}{n-1}-\int \tan^{n-2}(x)dx$, for $n\neq 1$.
5. Use the reduction formula above to compute $\displaystyle\int(\ln x)^3dx$ (so that there is no integral symbol at the end).
6. Try to come up with a reduction / induction formula for $$
I_n = \int \frac{1}{(1+x^3)^n} dx
$$by mimicking the example from in class / notes. If you are successful, it should look something like this : $I_{n+1} =\text{something} +\text{something}\cdot I_n$. (It should look quite similar to the example from class.)
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