Integrating Term By Term, Evaluate The Integral:$\[ \int \frac{5x^3 - 7x + 1}{x^3 - X^2} \, Dx \\]

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Introduction

In calculus, integration is a fundamental concept that deals with finding the area under curves, volumes of solids, and other mathematical quantities. One of the most common methods of integration is the term-by-term method, which involves breaking down the integrand into simpler components and integrating each term separately. In this article, we will explore the term-by-term method of integration and apply it to evaluate the integral ∫5x3−7x+1x3−x2 dx\int \frac{5x^3 - 7x + 1}{x^3 - x^2} \, dx.

Understanding the Term-by-Term Method

The term-by-term method of integration is a powerful technique that allows us to break down complex integrals into simpler components. This method involves the following steps:

  1. Break down the integrand: The first step is to break down the integrand into simpler components. This can be done by factoring the numerator and denominator, or by using other algebraic manipulations.
  2. Identify the terms: Once the integrand has been broken down, we need to identify the individual terms that make up the integral.
  3. Integrate each term: The final step is to integrate each term separately using the appropriate integration rules.

Evaluating the Integral

Now that we have a good understanding of the term-by-term method, let's apply it to evaluate the integral ∫5x3−7x+1x3−x2 dx\int \frac{5x^3 - 7x + 1}{x^3 - x^2} \, dx.

Step 1: Break down the integrand

The first step is to break down the integrand into simpler components. We can do this by factoring the numerator and denominator:

5x3−7x+1x3−x2=5x3x3−x2−7xx3−x2+1x3−x2\frac{5x^3 - 7x + 1}{x^3 - x^2} = \frac{5x^3}{x^3 - x^2} - \frac{7x}{x^3 - x^2} + \frac{1}{x^3 - x^2}

Step 2: Identify the terms

Now that we have broken down the integrand, we need to identify the individual terms that make up the integral. In this case, we have three terms:

  1. 5x3x3−x2\frac{5x^3}{x^3 - x^2}
  2. −7xx3−x2-\frac{7x}{x^3 - x^2}
  3. 1x3−x2\frac{1}{x^3 - x^2}

Step 3: Integrate each term

The final step is to integrate each term separately using the appropriate integration rules. Let's start with the first term:

∫5x3x3−x2 dx\int \frac{5x^3}{x^3 - x^2} \, dx

We can use the substitution method to integrate this term. Let u=x3−x2u = x^3 - x^2, then du=(3x2−2x) dxdu = (3x^2 - 2x) \, dx. Substituting these values into the integral, we get:

∫5x3x3−x2 dx=∫5uu⋅13x2−2x du\int \frac{5x^3}{x^3 - x^2} \, dx = \int \frac{5u}{u} \cdot \frac{1}{3x^2 - 2x} \, du

Simplifying the integral, we get:

∫5x3x3−x2 dx=∫53x2−2x du\int \frac{5x^3}{x^3 - x^2} \, dx = \int \frac{5}{3x^2 - 2x} \, du

Now, we can integrate this term using the standard integration rules:

∫53x2−2x du=52ln⁡∣3x2−2x∣+C\int \frac{5}{3x^2 - 2x} \, du = \frac{5}{2} \ln |3x^2 - 2x| + C

where CC is the constant of integration.

Step 4: Integrate the remaining terms

Now that we have integrated the first term, let's move on to the remaining terms. The second term is:

∫−7xx3−x2 dx\int -\frac{7x}{x^3 - x^2} \, dx

We can use the substitution method to integrate this term. Let u=x3−x2u = x^3 - x^2, then du=(3x2−2x) dxdu = (3x^2 - 2x) \, dx. Substituting these values into the integral, we get:

∫−7xx3−x2 dx=∫−7uu⋅13x2−2x du\int -\frac{7x}{x^3 - x^2} \, dx = \int -\frac{7u}{u} \cdot \frac{1}{3x^2 - 2x} \, du

Simplifying the integral, we get:

∫−7xx3−x2 dx=∫−73x2−2x du\int -\frac{7x}{x^3 - x^2} \, dx = \int -\frac{7}{3x^2 - 2x} \, du

Now, we can integrate this term using the standard integration rules:

∫−73x2−2x du=−72ln⁡∣3x2−2x∣+C\int -\frac{7}{3x^2 - 2x} \, du = -\frac{7}{2} \ln |3x^2 - 2x| + C

where CC is the constant of integration.

The final term is:

∫1x3−x2 dx\int \frac{1}{x^3 - x^2} \, dx

We can use the substitution method to integrate this term. Let u=x3−x2u = x^3 - x^2, then du=(3x2−2x) dxdu = (3x^2 - 2x) \, dx. Substituting these values into the integral, we get:

∫1x3−x2 dx=∫1u⋅13x2−2x du\int \frac{1}{x^3 - x^2} \, dx = \int \frac{1}{u} \cdot \frac{1}{3x^2 - 2x} \, du

Simplifying the integral, we get:

∫1x3−x2 dx=∫13x2−2x du\int \frac{1}{x^3 - x^2} \, dx = \int \frac{1}{3x^2 - 2x} \, du

Now, we can integrate this term using the standard integration rules:

∫13x2−2x du=12ln⁡∣3x2−2x∣+C\int \frac{1}{3x^2 - 2x} \, du = \frac{1}{2} \ln |3x^2 - 2x| + C

where CC is the constant of integration.

Step 5: Combine the results

Now that we have integrated each term separately, we can combine the results to get the final answer:

∫5x3−7x+1x3−x2 dx=52ln⁡∣3x2−2x∣−72ln⁡∣3x2−2x∣+12ln⁡∣3x2−2x∣+C\int \frac{5x^3 - 7x + 1}{x^3 - x^2} \, dx = \frac{5}{2} \ln |3x^2 - 2x| - \frac{7}{2} \ln |3x^2 - 2x| + \frac{1}{2} \ln |3x^2 - 2x| + C

Simplifying the expression, we get:

∫5x3−7x+1x3−x2 dx=12ln⁡∣3x2−2x∣+C\int \frac{5x^3 - 7x + 1}{x^3 - x^2} \, dx = \frac{1}{2} \ln |3x^2 - 2x| + C

where CC is the constant of integration.

Conclusion

Introduction

In our previous article, we explored the term-by-term method of integration and applied it to evaluate the integral ∫5x3−7x+1x3−x2 dx\int \frac{5x^3 - 7x + 1}{x^3 - x^2} \, dx. In this article, we will answer some of the most frequently asked questions about term-by-term integration.

Q: What is term-by-term integration?

A: Term-by-term integration is a method of integration that involves breaking down the integrand into simpler components and integrating each term separately.

Q: When can I use term-by-term integration?

A: You can use term-by-term integration when the integrand can be broken down into simpler components, such as fractions, products, or quotients.

Q: How do I break down the integrand into simpler components?

A: To break down the integrand into simpler components, you can use algebraic manipulations such as factoring, canceling, or combining like terms.

Q: What are some common mistakes to avoid when using term-by-term integration?

A: Some common mistakes to avoid when using term-by-term integration include:

  • Not breaking down the integrand into simpler components
  • Not identifying the individual terms
  • Not integrating each term separately
  • Not combining the results correctly

Q: Can I use term-by-term integration with trigonometric functions?

A: Yes, you can use term-by-term integration with trigonometric functions. However, you may need to use trigonometric identities to simplify the integrand.

Q: Can I use term-by-term integration with exponential functions?

A: Yes, you can use term-by-term integration with exponential functions. However, you may need to use properties of exponents to simplify the integrand.

Q: Can I use term-by-term integration with logarithmic functions?

A: Yes, you can use term-by-term integration with logarithmic functions. However, you may need to use properties of logarithms to simplify the integrand.

Q: What are some examples of term-by-term integration?

A: Some examples of term-by-term integration include:

  • ∫5x3−7x+1x3−x2 dx\int \frac{5x^3 - 7x + 1}{x^3 - x^2} \, dx
  • ∫2x2+3x−1x2+2x dx\int \frac{2x^2 + 3x - 1}{x^2 + 2x} \, dx
  • ∫3x2−2x+1x2−x dx\int \frac{3x^2 - 2x + 1}{x^2 - x} \, dx

Q: How do I evaluate the integral of a rational function using term-by-term integration?

A: To evaluate the integral of a rational function using term-by-term integration, you can follow these steps:

  1. Break down the integrand into simpler components
  2. Identify the individual terms
  3. Integrate each term separately
  4. Combine the results correctly

Q: How do I evaluate the integral of a trigonometric function using term-by-term integration?

A: To evaluate the integral of a trigonometric function using term-by-term integration, you can follow these steps:

  1. Break down the integrand into simpler components
  2. Identify the individual terms
  3. Use trigonometric identities to simplify the integrand
  4. Integrate each term separately
  5. Combine the results correctly

Q: How do I evaluate the integral of an exponential function using term-by-term integration?

A: To evaluate the integral of an exponential function using term-by-term integration, you can follow these steps:

  1. Break down the integrand into simpler components
  2. Identify the individual terms
  3. Use properties of exponents to simplify the integrand
  4. Integrate each term separately
  5. Combine the results correctly

Q: How do I evaluate the integral of a logarithmic function using term-by-term integration?

A: To evaluate the integral of a logarithmic function using term-by-term integration, you can follow these steps:

  1. Break down the integrand into simpler components
  2. Identify the individual terms
  3. Use properties of logarithms to simplify the integrand
  4. Integrate each term separately
  5. Combine the results correctly

Conclusion

In this article, we have answered some of the most frequently asked questions about term-by-term integration. We have discussed when to use term-by-term integration, how to break down the integrand into simpler components, and how to avoid common mistakes. We have also provided examples of term-by-term integration and discussed how to evaluate the integral of various types of functions using this method.