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50 6.7 Integration By Substitution Homework

Integration By Substitution Worksheet —
Integration By Substitution Worksheet — from db-excel.com

Introduction

Welcome to today's blog article on 6.7 integration by substitution homework. In this article, we will explore the concept of integration by substitution and provide you with some helpful tips and examples to assist you with your homework. Integration by substitution is a powerful technique used to solve integrals that involve a composition of functions. By substituting a new variable, we can simplify the integral and make it easier to solve. So let's dive in and learn more about this topic.

1. Understanding Integration by Substitution

In this section, we will provide a brief overview of integration by substitution. Integration by substitution, also known as u-substitution, is a method used to simplify integrals by introducing a new variable. This new variable is chosen in such a way that it simplifies the integrand and makes the integral easier to evaluate. The basic idea behind integration by substitution is to replace the original variable with a new variable that relates to the original variable through a known function.

2. Choosing the Right Substitution

Choosing the right substitution is crucial for successfully solving integrals using the method of substitution. In general, we look for a substitution that simplifies the integrand and reduces it to a known form. The choice of substitution depends on the structure of the integrand and requires some intuition and practice. Some common substitutions include:

  • Trigonometric substitutions
  • Exponential substitutions
  • Logarithmic substitutions
  • Hyperbolic substitutions
  • Algebraic substitutions

3. Step-by-Step Process

Now let's walk through the step-by-step process of solving integrals using the method of substitution:

  1. Identify the part of the integrand that can be simplified by substitution.
  2. Choose a new variable u to substitute for the simplified part.
  3. Compute the derivative du/dx of the new variable u.
  4. Rewrite the integral in terms of the new variable u and du.
  5. Solve the new integral with respect to u.
  6. Replace the new variable u with the original variable x in the final answer.

4. Example Problem 1

Let's work through an example problem to illustrate the process of integration by substitution:

Example: Evaluate the integral ∫ 2x cos(x^2) dx.

  1. Identify the part of the integrand that can be simplified: cos(x^2).
  2. Choose a new variable u to substitute for the simplified part: u = x^2.
  3. Compute the derivative du/dx: du/dx = 2x.
  4. Rewrite the integral in terms of u and du: ∫ cos(u) du.
  5. Solve the new integral with respect to u: ∫ cos(u) du = sin(u) + C.
  6. Replace u with x^2 in the final answer: sin(x^2) + C.

5. Example Problem 2

Let's try another example problem to further strengthen our understanding:

Example: Evaluate the integral ∫ (3x^2 + 2x + 1) e^(x^3 + x^2 + x) dx.

  1. Identify the part of the integrand that can be simplified: e^(x^3 + x^2 + x).
  2. Choose a new variable u to substitute for the simplified part: u = x^3 + x^2 + x.
  3. Compute the derivative du/dx: du/dx = 3x^2 + 2x + 1.
  4. Rewrite the integral in terms of u and du: ∫ e^u du.
  5. Solve the new integral with respect to u: ∫ e^u du = e^u + C.
  6. Replace u with x^3 + x^2 + x in the final answer: e^(x^3 + x^2 + x) + C.

6. Tips for Success

Here are some helpful tips to keep in mind when working on integration by substitution problems:

  • Practice identifying the part of the integrand that can be simplified.
  • Choose substitutions that simplify the integrand and reduce it to a known form.
  • Double-check your work by differentiating the final answer to ensure it matches the original integrand.
  • Work on a variety of problems to gain a deeper understanding of the method of substitution.
  • Seek additional resources, such as textbooks or online tutorials, to further enhance your understanding of integration by substitution.

7. Conclusion

Integration by substitution is a powerful technique that allows us to simplify integrals and make them easier to solve. By choosing the right substitution, we can transform a complex integral into a more manageable form. In this article, we explored the concept of integration by substitution, discussed the step-by-step process, and worked through example problems. We also provided some helpful tips to assist you with your integration by substitution homework. Remember to practice regularly and seek additional resources if needed. Happy integrating!