Find an anti derivative (or integral) of the following functions by the method of inspection.

1. \(\int{sin{2}xdx}\)

2. \(\int{cos{3}xdx}\)

3. \(\int{e^{2x}dx}\)

4. \(\int{\left(ax+b\right)^2dx}\)

5. \(\int{sin{2}x-4e^{3x}dx}\)

Find the following integrals in Exercises 6 to 20:

6. \(\int\left(4e^{3x}+1\right)dx\)

7. \(\int{x^2\left(1-\frac{1}{x^2}\right)dx}\)

8. \(\int\left(ax^2+bx+c\right)dx\)

9. \(\int\left(2x^2+e^x\right)dx\)

10. \(\int{\left(\sqrt x-\frac{1}{\sqrt x}\right)^2dx}\)

11. \(\int{\frac{x^3+5x^2-4}{x^2}dx}\)

12. \(\int{\frac{x^3+3x+4}{\sqrt x}dx}\)

13. \(\int{\frac{x^3-x^2+x-1}{x-1}dx}\)

14. \(\int{\left(1-x\right)\sqrt x\ dx}\)

15. \(\int{\sqrt x\left(3x^2+2x+3\right)dx}\)

16. \(\int\left(2x-3cos{x}+e^x\right)dx\)

17. \(\int\left(2x^2-3sin{x}+5\sqrt x\right)dx\)

18. \(\int{sec{x}\left(sec{x}+tan{x}\right)dx}\)

19. \(\int{\frac{{sec}^2{x}}{cos{e}c^2x}dx}\)

20. \(\int{\frac{2-3sin{x}}{{cos}^2{x}}dx}\)

Choose the correct answer in Exercises 21 and 22.

21. The anti derivative of \(\left(\sqrt x+\frac{1}{\sqrt x}\right)\) equals
(A) \(\frac{1}{3}x^\frac{1}{3}+2x^\frac{1}{2}+C\)
(B) \(\frac{2}{3}x^\frac{2}{3}+\frac{1}{2}x^2+C\)
(C) \(\frac{2}{3}x^\frac{3}{2}+2x^\frac{1}{2}+C\)
(D) \(\frac{3}{2}x^\frac{3}{2}+\frac{1}{2}x^\frac{1}{2}+C\)

22. If \(\frac{d}{dx}f\left(x\right)=4x^3-\frac{3}{x^4}\) such that \(f\left(2\right)=0\). Then \(f\left(x\right)\) is
(A) \(x^4+\frac{1}{x^3}-\frac{129}{8}\)
(B) \(x^3+\frac{1}{x^4}+\frac{129}{8}\)
(C) \(x^4+\frac{1}{x^3}+\frac{129}{8}\)
(D) \(x^3+\frac{1}{x^4}-\frac{129}{8}\)