https://file.aiquickdraw.com/mathbot/file/17289247298234rl978nt.jpg

I see that you've uploaded an image containing several integrals with their solutions. I will break them down for you if you'd like, or you can ask me about any specific problem for further explanation.

Here’s a quick breakdown of the integrals shown in the image:

1. \( \int \frac{\log x}{x} \, dx \)
   **Answer**: \( \frac{(\log x)^2}{2} + C \)

2. \( \int x e^{-x} \, dx \)
   **Answer**: \( -xe^{-x} - e^{-x} + C \)

3. \( \int (e^{3x} - 7^x + 6) \, dx \)
   **Answer**: \( \frac{e^{3x}}{3} - \frac{7^x}{\log 7} + 6x + C \)

4. \( \int \frac{(6x - 5)}{(3x^2 - 5x + 1)} \, dx \)
   **Answer**: \( \frac{2}{3} \log(3x^2 - 5x + 1) + C \)

5. \( \int x^2 e^{3x} \, dx \)
   **Answer**: \( \frac{x^2 e^{3x}}{3} - \frac{2xe^{3x}}{9} + \frac{2e^{3x}}{27} + C \)

6. \( \int \frac{1}{ax + b} \, dx \)
   **Answer**: \( \frac{1}{a} \log(ax + b) + C \)

7. \( \int \frac{2ax + b}{ax^2 + bx + c} \, dx \)
   **Answer**: \( \log(ax^2 + bx + c) + C \)

8. \( \int \left( x^{1/3} + x(x^{2/3} + 1) \right) \, dx \)
   **Answer**: \( \frac{3}{2} x^{4/3} + \frac{1}{10} (x^2 + 1)^5 + C \)

9. \( \int \frac{6x + 5}{x^2 - 3x} \, dx \)
   **Answer**: \( 6x + \frac{23}{2\sqrt{3}} \log\left( \frac{x - \sqrt{3}}{x + \sqrt{3}} \right) + C \)

10. \( \int \frac{x}{x^2 - 3x + 2} \, dx \)
    **Answer**: \( -\log(x - 1) + 2 \log(x - 2) + C \)

**Definite Integrals**:

11. \( \int_0^3 (16 - 2x + 3x^2) \, dx \)
    **Answer**: 50

12. \( \int_0^5 (35 - 3x - x^2) \, dx \)
    **Answer**: 107.83

13. \( \int_0^a (x^2 - ax + a^2) \, dx \)
    **Answer**: \( \frac{5}{6} a^3 \)

Feel free to ask for more details on any of these integrals or let me know if you would like to go through the solutions step by step.

Would you like more details on any specific integral? Also, here are some relative questions for further practice:

1. Can you explain the technique used in solving integral (7)?
2. What is the relationship between the logarithmic and exponential integrals?
3. How do you handle integrals involving polynomials and square roots, such as in problem (4)?
4. Could you clarify the steps involved in solving definite integrals like (12)?
5. How does integration by parts help in problem (5)?

**Tip**: When dealing with logarithmic integrals, always check if you can simplify the problem using substitution, especially when the denominator is linear.

Solve the following integrals and definite integrals.

This page provides solutions to various integrals and definite integrals, including log and exponential functions, polynomial integrals, and more advanced calculus concepts. Suitable for high school seniors and college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation