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# Week 13 Problem Set B.
## Reading.
Chapter 5.4, 5.5, 5.6
On indefinite integrals and the substitution method.
## Problems.
### Evaluating indefinite integrals.
Evaluate the following indefinite integrals. You should try to find an appropriate substitution to help you. You may need some trigonometric identities and algebraic manipulations sometimes. And don't forget chain rule. You should do a lot of these for practice.
1. $\displaystyle \int 2(2x + 4)^{5}dx$
2. $\displaystyle \int 2x (x^{2} + 5)^{-4} dx$
3. $\displaystyle \int \frac{4x^{3}}{(x^{4}+1)^{2}}dx$
4. $\displaystyle \int (3x + 2)(3x^{2} + 4x)^{4}dx$
5. $\displaystyle \int \frac{(1+\sqrt{x})^{1 /3}}{\sqrt{x}}dx$
6. $\displaystyle \int x \sin(2x^{2}) dx$
7. $\displaystyle \int \sec(2x)\tan(2x)dx$
8. $\displaystyle \int \left( 1-\cos \frac{x}{2} \right)^{2} \sin \frac{x}{2} \,dx$
9. $\displaystyle \int \frac{9x^{2} dx}{\sqrt{1-x^{3}}}$
10. $\displaystyle \int \sqrt{x} \sin^{2}(x^{3 / 2} - 1) dx$
11. $\displaystyle\int \frac{1}{x^{2}} \cos^{2}(\frac{1}{x})dx$
12. $\displaystyle \int \frac{dx}{\sqrt{5x + 8}}$
13. $\displaystyle \int \theta \sqrt[4]{1-\theta^{2}} d\theta$
14. $\displaystyle\int \frac{1}{\sqrt{x}(1+\sqrt{x})^{2}} dx$
15. $\displaystyle\int \sec^{2}(3x+2) dx$
16. $\displaystyle\int \sin^{5} \frac{x}{3} \cos \frac{x}{3} dx$
17. $\displaystyle \int r^{2} \left( \frac{r^{3}}{18}-1 \right)^{5}dr$
18. $\displaystyle\int x^{1 / 2} \sin(x^{3 / 2}+ 1)dx$
19. $\displaystyle \int \csc\left( \frac{v - \pi}{2} \right) \cot\left( \frac{v-\pi}{2} \right) dv$
20. $\displaystyle \int \frac{\sin(2t+1)}{\cos^{2}(2t+1)}dt$
21. $\displaystyle \int \frac{\sec z \tan z}{\sqrt{\sec z}}dz$
22. $\displaystyle \int \frac{1}{t^{2}} \cos \left( \frac{1}{t}-1 \right)dt$
23. $\displaystyle \int \frac{1}{\theta^{2}}\sin \frac{1}{\theta}\cos \frac{1}{\theta}d\theta$
24. $\displaystyle\int \sqrt{\frac{x-1}{x^{5}}}dx$ .... This one may be tricky, but doable using substitution, after some appropriate algebraic manipulation.
25. $\displaystyle \int \frac{1}{x^{2}} \sqrt{2 - \frac{1}{x}}dx$
26. $\displaystyle \int \frac{1}{x^{3}} \sqrt{\frac{x^{2}-1}{x^{2}}}dx$
27. $\displaystyle \int \sqrt{\frac{x^{3}-3}{x^{11}}} dx$
28. $\displaystyle\int \sqrt{\frac{x^{4}}{x^{3}-1}}dx$
29. $\displaystyle \int x(x-1)^{10}dx$
30. $\displaystyle\int x \sqrt{4-x} \,dx$
31. $\displaystyle\int x^{3} \sqrt{x^{2}+1}\, dx$
32. $\displaystyle \int(x+5)(x-5)^{1 / 3} dx$
33. $\displaystyle \int 3 x^{5} \sqrt{x^{3}+1}dx$
34. $\displaystyle \int \frac{x}{(x^{2}-4)^{3}}dx$
35. $\displaystyle\int \frac{x}{(x-4)^{3}}dx$
When it is not clear what substitution to make, you could try to reduce the problem step by step, using substitutions to reduce the complexity of the expression one by one. Remember to keep track of chain rule.
1. Compute $\displaystyle \int \frac{18\tan^{2}x \sec^{2}x}{(2+\tan^{3}x)^{2}} dx$ by the substitution $u = \tan x$, followed by $v = u^{3}$, then followed by $w=2+v$.
2. Compute $\displaystyle \int \sqrt{1+\sin^{2}(x-1)}\sin(x-1)\cos(x-1)dx$ by the substitution $u=x-1$, followed by $v = \sin(u)$, then by $w=1+v^{2}$.
3. Compute the following, you may want to try a sequence of substitutions like before:
1. $$
\int \frac{(2r-1) \cos \sqrt{3(2r-1)^2+6}}{\sqrt{3(2r-1)^{2}+6}}dr
$$
2. $$
\int \frac{\sin \sqrt{\theta}}{\sqrt{\theta \cos^{3}\sqrt{\theta}}} d\theta
$$
### Initial value problems.
For each of the following, solve for a function that satisfy the given conditions:
1. $\displaystyle \frac{ds}{dt} = 12t (3t^{2}-1)^{3}$, $s(1)=3$.
2. $\displaystyle \frac{dy}{dx} = 4x (x^{2}+8)^{-1/3}$, $y(0) = 0$.
3. $\displaystyle \frac{ds}{dt}= 8 \sin^{2}(t+ \frac{\pi}{12})$, $s(0) = 8$.
4. $\displaystyle \frac{d^{2}s}{dt^2} = -4 \sin(2t - \frac{\pi}{2})$, $s'(0) = 100$, $s(0) = 0$.
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