# BC Calculus: Assignment 48: Review for Chapter 6 Test Worksheet

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

In the following problems, find $$f'(x)$$.
1. $$f(x) = \ln{(x^4 + 8)}$$
2. Solution $$f'(x) = \frac{4x^3}{x^4 + 8}$$
3. $$f(x) = \ln{(3x\sqrt{3 - x})}$$
4. Solution $$f(x) = \ln{3} + \ln{x} + \frac{1}{2} \ln{(3 - x)} \implies f'(x) = \frac{1}{x} - \frac{1}{2(3 - x)}$$
5. $$f(x) = \ln{ \left( \frac{5x^2}{\sqrt{5 + x^2}} \right) }$$
6. Solution $$f(x) = \ln{5} + 2\ln{x} - \frac{1}{2} \ln{(5 + x^2)} \implies f'(x) = \frac{2}{x} - \frac{2x}{2(5 + x^2)}$$
7. $$f(x) = e^{x \cos{x}}$$
8. Solution $$f'(x) = e^{x \cos{x}} ( \cos{x} - x \sin{x})$$
9. $$f(x) = e^{-3x}\sin{(5x)}$$
10. Solution $$f'(x) = e^{-3x}\cos{(5x)}5 + \sin{(5x)} e^{-3x}(-3)$$
11. $$f(x) = e^{\ln{\frac{1}{x}}}$$
12. Solution $$f(x) = \frac{1}{x} = x^{-1} \implies f'(x) = -x^{-2}$$
13. $$f(x) = log_{12} { \left( x^3 \right) }$$
14. Solution $$f'(x) = \frac{3}{x \ln{12}}$$
15. $$f(x) = x^5 5^x$$
16. Solution $$f'(x) = x^55^x\ln{5} + 5^x5x^4$$
Evaluate the following integrals:
1. $$\displaystyle \int\frac{\sec^2{x}}{\tan{x}} \, dx$$
2. Solution $$\ln{|\tan{x}|} + C$$
3. $$\displaystyle \int\frac{\cos{x}}{1 - \sin{x}} \, dx$$
4. Solution $$-\ln{|1 - \sin{x}|} + C$$
5. $$\displaystyle \int\frac{1}{x \ln{x}} \, dx$$
6. Solution $$\ln {|\ln{x}|} + C$$
7. $$\displaystyle \int\frac{1}{x} \cos{( \ln{x} )} \, dx$$
8. Solution $$\sin{(\ln{x})} + C$$
9. $$\displaystyle \int\frac{\sin{x} - \cos{x}}{\cos{x}} \, dx$$
10. Solution $$\ln{|\sec{x}|} - x + C$$
11. $$\displaystyle \int\frac{dx}{\sqrt{x} \left( 1 + 2\sqrt{x} \right)}$$
12. Solution $$\ln{|1 + 2\sqrt{x}|} + C$$
13. $$\displaystyle \int x \cdot 4^{-x^2} \, dx$$
14. Solution $$-\frac{4^{-x^2}}{2\ln{4}} + C$$
15. $$\displaystyle \int 7^{\sin{x}}\cos{x} \, dx$$
16. Solution $$\frac{7^{\sin{x}}}{\ln{7}} + C$$