Замена переменных и внесенние под дифференциал в неопределенном интеграле

Вычислите неопределенные интегралы:

1. \( \displaystyle \int \sin(3x) \, dx \)
2. \( \displaystyle \int e^{2t} \, dt \)
3. \( \displaystyle \int 5\cos(4x) \, dx \)
4. \( \displaystyle \int \sqrt{x} \, dx \)
5. \( \displaystyle \int \sec^2(x) \, dx \)
6. \( \displaystyle \int \frac{\cos x}{\sin^2 x} \, dx \)
7. \(* \displaystyle \int \frac{1}{1 + e^t} \, dt \)
8. \( \displaystyle \int \frac{1}{1+4x^2} \, dx \)
9. \( \displaystyle \int \frac{\cos(x)}{\sin^2(x)} \, dx \)
10. \( \displaystyle \int \frac{1}{\sqrt{1 - 16x^2}} \, dx \)
11. \( \displaystyle \int \frac{e^t}{1 + e^t} \, dt \)
12. \( \displaystyle \int \frac{\sin(x)}{\cos^2(x)} \, dx \)
13. \( \displaystyle \int \frac{1}{\sqrt{a^2 - x^2}} \, dx \), где \( a \) - константа
14. \( \displaystyle \int \frac{2x}{(x^2 + 1)^2} \, dx \)
15. \( \displaystyle \int \frac{3}{\sqrt[3]{t^2}} \, dt \)
16. \( \displaystyle \int \frac{\operatorname{tg}(x)}{\cos(x)} \, dx \)
17. \( \displaystyle \int \frac{e^t}{(e^{2t}+1)^2} \, dt \)
18. \( \displaystyle \int \frac{\arctan(x)}{1+x^2} \, dx \)
19. \( \displaystyle \int \frac{\sec(x)}{\tan(x)} \, dx \)
20. \( \displaystyle \int \frac{\sin(3t)}{\cos^2(3t)} \, dt \)
21. \( \displaystyle \int \frac{\arcsin(x)}{\sqrt{1-x^2}} \, dx \)
22. \( \displaystyle \int \frac{2}{\sqrt{4t^2-1}} \, dt \)
23. \( \displaystyle \int \frac{\tan(x)}{\sqrt{\cos(x)}} \, dx \)
24. \( \displaystyle \int \frac{e^t}{\sqrt{1 - e^{2t}}} \, dt \)
25. \( \displaystyle \int \frac{1}{(1 + x)^2} \, dx \)
26. \( \displaystyle \int \frac{\sin(t)}{\sqrt{1 + \cos^2(t)}} \, dt \)
27. \( \displaystyle \int \frac{1}{\sqrt[3]{x^2}} \, dx \)
28. \( \displaystyle \int \frac{e^t}{\sqrt{e^{2t} - 1}} \, dt \)
29. \( \displaystyle \int \frac{\sqrt{2 + \ln(t)}}{t} \, dt \)
30. \( \displaystyle \int \frac{e^t}{(e^t + 1)^2} \, dt \)
31. \( \displaystyle \int \frac{\sqrt{1 + e^x}}{e^x} \, dx \)