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Let's solve the integrals one by one:

1. \( \int dx \)
   - Solution: \( x + C \)

2. \( \int \frac{dx}{x} \)
   - Solution: \( \ln |x| + C \)

3. \( \int x^{3/4} dx \)
   - Solution: \( \frac{4}{7} x^{7/4} + C \)

4. \( \int 5x^3 dx \)
   - Solution: \( \frac{5x^4}{4} + C \)

5. \( \int 2bx^3 dx \)
   - Solution: \( \frac{2b}{4} x^4 + C = \frac{b}{2} x^4 + C \)

6. \( \int \left(x^4 - x^2 + \frac{1}{x^3} - \frac{1}{x^2}\right) dx \)
   - Solution: 
     - \( \int x^4 dx = \frac{x^5}{5} \)
     - \( \int -x^2 dx = -\frac{x^3}{3} \)
     - \( \int \frac{1}{x^3} dx = -\frac{1}{2x^2} \)
     - \( \int -\frac{1}{x^2} dx = \frac{1}{x} \)
     - Combined: \( \frac{x^5}{5} - \frac{x^3}{3} - \frac{1}{2x^2} + \frac{1}{x} + C \)

7. \( \int 5(5x - 1)^3 dx \)
   - Use substitution: Let \( u = 5x - 1 \), then \( du = 5 dx \).
   - Integral becomes: \( \int u^3 du = \frac{u^4}{4} \)
   - Solution: \( \frac{(5x - 1)^4}{4} + C \)

8. \( \int \sqrt[4]{x} dx \)
   - Rewrite \( \sqrt[4]{x} = x^{1/4} \)
   - Solution: \( \frac{4}{5} x^{5/4} + C \)

9. \( \int \frac{dx}{(x - 1)^5} \)
   - Use substitution: Let \( u = x - 1 \), then \( du = dx \).
   - Integral becomes: \( \int u^{-5} du = \frac{u^{-4}}{-4} \)
   - Solution: \( -\frac{1}{4(x - 1)^4} + C \)

10. \( \int 3\sqrt{x^2} dx \)
    - Simplify: \( \sqrt{x^2} = |x| \)
    - Integral becomes: \( \int 3|x| dx \)
    - Split into two cases:
      - For \( x \geq 0 \), \( |x| = x \), and \( \int 3x dx = \frac{3x^2}{2} \)
      - For \( x < 0 \), \( |x| = -x \), and \( \int 3(-x) dx = -\frac{3x^2}{2} \)
    - Solution: \( \frac{3x^2}{2} + C \) for \( x \geq 0 \) or \( -\frac{3x^2}{2} + C \) for \( x < 0 \)

Would you like further details on any of these integrals?

Here are 5 related questions:
1. How do you solve integrals with polynomial functions?
2. What is the substitution method for integrals?
3. How do you deal with absolute values in integrals?
4. What is the general formula for integrating powers of \( x \)?
5. How can partial fractions be used in integration?

**Tip**: For integrals involving \( x^n \), where \( n \neq -1 \), the general formula is \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \).

Solve the following integrals: ∫dx, ∫dx/x, ∫x^(3/4)dx, ∫5x^3 dx, ∫2bx^3dx, ∫(x^4 - x^2 + 1/x^3 - 1/x^2) dx, ∫5(5x - 1)^3 dx, ∫4√x dx, ∫dx/(x - 1)^5, ∫3√x^2dx

Solve advanced integrals step-by-step, including polynomial and rational functions, using techniques such as the power rule and substitution. This includes integrals like ∫x^(3/4)dx, ∫5x^3dx, ∫(x^4 - x^2 + 1/x^3 - 1/x^2)dx, and more. Suitable for university-level calculus students.

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