Indefinite integrals. Determine the following indefinite integrals. Check your work by differentiation.
(Question 31 and 35) Show all work, thank you!
Answers are given.
Transcribed Image Text:### Integral Exercises
Here are several integral exercises designed to test and improve your integration skills. Try to solve these integrals and understand the methods used in each step.
#### Exercise 30
\[
\int 6 \sqrt[3]{x} \, dx
\]
#### Exercise 31
\[
\int (3x + 1)(4 - x) \, dx
\]
#### Exercise 32
\[
\int (4z^{1/3} - z^{-1/3}) \, dz
\]
#### Exercise 33
\[
\int \left( \frac{3}{x^4} + 2 - \frac{3}{x^2} \right) \, dx
\]
#### Exercise 34
\[
\int \sqrt[5]{r^2} \, dr
\]
#### Exercise 35
\[
\int \frac{4x^4 - 6x^2}{x} \, dx
\]
#### Exercise 36
\[
\int \frac{12t^8 - t}{t^{3/2}} \, dt
\]
#### Exercise 37
\[
\int \frac{x^2 - 36}{x - 6} \, dx
\]
### Explanation of Integrals:
- **Exercise 30:** Integrate a cube root function multiplied by a constant.
- **Exercise 31:** Expand the product of polynomials and integrate the resulting expression.
- **Exercise 32:** Integrate a function containing different powers of \( z \).
- **Exercise 33:** Integrate a rational function consisting of multiple terms.
- **Exercise 34:** Integrate a root function with a power of \( r \).
- **Exercise 35:** Simplify the rational expression by dividing polynomials before integrating.
- **Exercise 36:** Simplify by reducing the fraction involving powers of \( t \) and perform the integration.
- **Exercise 37:** Perform polynomial division if needed before integrating the rational function.
These exercises cover a range of techniques, including polynomial expansion, simplification, and rational function integration. Working through these problems will improve your ability to integrate a variety of functions.
Transcribed Image Text:### Integral Calculus Problems
Here is a set of integral equations that can serve as practice problems. Each expression is followed by a constant \( C \) indicating the integration constant.
29. \(\int \frac{9}{4} x^{4/3} + 6x^{2/3} + 6x \, dx + C\)
31. \(\int -x^3 + \frac{11}{2} x^2 + 4x \, dx + C\)
33. \(\int -x^{-3} + 2x + 3x^{-1} \, dx + C\)
35. \(\int x^4 - 3x^2 \, dx + C\)
37. \(\int \frac{1}{2} x^2 + 6x \, dx + C\)
These problems involve integrating polynomial and rational functions. The solutions will involve applying standard integration techniques for each term and ensuring the integration constant \( C \) is included.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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![### Integral Exercises
Here are several integral exercises designed to test and improve your integration skills. Try to solve these integrals and understand the methods used in each step.
#### Exercise 30
\[
\int 6 \sqrt[3]{x} \, dx
\]
#### Exercise 31
\[
\int (3x + 1)(4 - x) \, dx
\]
#### Exercise 32
\[
\int (4z^{1/3} - z^{-1/3}) \, dz
\]
#### Exercise 33
\[
\int \left( \frac{3}{x^4} + 2 - \frac{3}{x^2} \right) \, dx
\]
#### Exercise 34
\[
\int \sqrt[5]{r^2} \, dr
\]
#### Exercise 35
\[
\int \frac{4x^4 - 6x^2}{x} \, dx
\]
#### Exercise 36
\[
\int \frac{12t^8 - t}{t^{3/2}} \, dt
\]
#### Exercise 37
\[
\int \frac{x^2 - 36}{x - 6} \, dx
\]
### Explanation of Integrals:
- **Exercise 30:** Integrate a cube root function multiplied by a constant.
- **Exercise 31:** Expand the product of polynomials and integrate the resulting expression.
- **Exercise 32:** Integrate a function containing different powers of \( z \).
- **Exercise 33:** Integrate a rational function consisting of multiple terms.
- **Exercise 34:** Integrate a root function with a power of \( r \).
- **Exercise 35:** Simplify the rational expression by dividing polynomials before integrating.
- **Exercise 36:** Simplify by reducing the fraction involving powers of \( t \) and perform the integration.
- **Exercise 37:** Perform polynomial division if needed before integrating the rational function.
These exercises cover a range of techniques, including polynomial expansion, simplification, and rational function integration. Working through these problems will improve your ability to integrate a variety of functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ea0963c-64ff-46a6-b97c-79944ef211ae%2Fe2150352-3a1a-4b83-af6c-ec543c5f6716%2Fpq2o96m_processed.jpeg&w=3840&q=75)
