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Homework 8 Problems
Section 8.3 problems
1. Find $\displaystyle\int \cos^3(x)\sin^2(x) \mathrm{d}x$.
2. Find $\displaystyle\int \sin^3(2\theta) \sqrt{\cos(2\theta)} \mathrm{d}\theta$.
3. Find $\displaystyle\int \sin^4(x)\cos^2(x)\mathrm{d}x$.
4. Find $\displaystyle\int \tan^5(x)\sec^2(x) \mathrm{d}x$.
5. Find $\displaystyle\int \sec^4(x) \mathrm{d}x$.
Section 8.4 problems
6. Find $\displaystyle\int \dfrac{4}{x^2\sqrt{16-x^2}} \mathrm{d}x$.
7. Find $\displaystyle\int \dfrac{1}{\sqrt{4+x^2}} \mathrm{d}x$.
8. Find $\displaystyle\int x\sqrt{x^2-36} \mathrm{d}x$.
9. Find $\displaystyle\int \dfrac{\sqrt{x^2-25}}{x} \mathrm{d}x$.
Honors section
1. Use integration by parts to verify the following formula for an arbitrary $n>1$ (i.e. keep "$n$" in your integrals!):
$$\displaystyle\int \cos^n(x) \mathrm{d}x = \dfrac{\cos^{n-1}(x)\sin(x)}{n} + \dfrac{n-1}{n} \displaystyle\int \cos^{n-2}(x) \mathrm{d}x.$$
2. Use the formula from the previous problem to establish the following formulas:
a.) if $n$ is odd and $n \geq 3$, then
$$\displaystyle\int_0^{\frac{\pi}{2}} \cos^n(x) \mathrm{d}x = \left( \dfrac{2}{3} \right) \left( \dfrac{4}{5} \right) \left( \dfrac{6}{7} \right) \ldots \left( \dfrac{n-1}{n} \right).$$
b.) if $n$ is even and $n \geq 2$, then
$$\displaystyle\int_0^{\frac{\pi}{2}} \cos^n(x) \mathrm{d}x = \left( \dfrac{1}{2} \right) \left( \dfrac{3}{4} \right) \left( \dfrac{5}{6} \right) \ldots \left( \dfrac{n-1}{n} \right) \left( \dfrac{\pi}{2} \right).$$
3. Recall that an ellipse obeys the formula $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}=1$ for some $a,b > 0$.
a.) Solve this equation for $y$ to get an equation for the "top half" of the ellipse.
b.) Integrate the function from part a.) over the interval $[-a,a]$ and multiply by 2 to find a formula for the area of the ellipse.
