# BC Calculus : §6.2 U-Sub Worksheet

• Class: 5H: BC Calculus
• Author: Peter Atlas
• Text: Calculus Finney, Demana, Waits, Kennedy

For each of the following, rewrite using mathematical notation, then evaluate without the use of a calculator.

1. $$\displaystyle \int_{0}^{3} (4 - t)^{-\frac{1}{2}} \, dt$$
2. Solution $$u = 4 - t$$, answer: 2
3. $$\displaystyle \int_{-1}^{0}(3u + 4)^{\frac{1}{2}} \, du$$
4. Solution $$v = 3u + 4$$, answer: $$\frac{2}{9} \left( 4^{\frac{3}{2}} - 1 \right)$$
5. $$\displaystyle \int_{2}^{3}(2y - 3)^{-1} \, dy$$
6. Solution $$u = 2y - 3$$, answer: $$\frac{\ln{3}} {2}$$
7. $$\displaystyle \int_{0}^{\sqrt{3}} x(4 - x^2)^{-\frac{1}{2}} \, dx$$
8. Solution $$u = 4 - x^2$$, answer: 1
9. $$\displaystyle \int_{1}^{2} \frac{3x - 1}{3x} \, dx$$
10. Solution Break it up into two integrals. Answer: $$1 - \frac{\ln{2}}{3}$$
11. $$\displaystyle \int_{0}^{1} e^{-x} \, dx$$
12. Solution $$u = -x$$, answer: $$1 - \frac{1}{e}$$
13. $$\displaystyle \int_{0}^{1} xe^{x^2} \, dx$$
14. Solution $$u = x^2$$, answer: $$\frac{e}{2} - \frac{1}{2}$$
15. $$\displaystyle \int_{1}^{e} \frac{\ln{x}}{x} \, dx$$
16. Solution $$u = \ln{x}$$, answer: $$\frac{1}{2}$$
17. $$\displaystyle \int_{0}^{\frac{\pi}{6}} \frac{ \cos{t}}{1 + 2 \sin{t}} \, dt$$
18. Solution $$u = 1 + 2 \sin{t}$$, answer: $$\frac{\ln{2}}{2}$$
19. $$\displaystyle \int_{0}^{1} \frac{e^{-x} +1}{e^{-x}} \, dx$$
20. Solution $$\displaystyle u = e^{-x}$$ then... grrr. I fell into my own trap. Break it up! Answer: $$e$$