Definite Integral - Introduction to the Method of Substitution
- The method of substitution is a technique used to evaluate definite integrals.
- It involves making a suitable substitution to simplify the integrand.
- This method is particularly useful when dealing with integrals that involve complicated algebraic or trigonometric expressions.
Steps for the Method of Substitution
- Identify a suitable substitution, typically denoted by u.
- Compute the derivative of u and substitute it in the integrand.
- Replace all occurrences of the original variable with u.
- Evaluate the resulting integral with respect to u.
- Finally, substitute back the original variable in terms of u to obtain the solution to the original integral.
Example 1
Evaluate the integral:
∫[(x^3 + 2x^2 - 1)/(x^2 + x + 1)] dx
- Let u = x^2 + x + 1
- Differentiating u with respect to x, we get du/dx = 2x + 1
- Replacing x^2 + x + 1 by u, the integral becomes:
∫[(x^3 + 2x^2 - 1)/u] du
- Now the integral can be evaluated using the laws of integration.
Example 2
Evaluate the integral:
∫[x*sin(x^2 + 1)] dx
- Let u = x^2 + 1
- Differentiating u with respect to x, we get du/dx = 2x
- Replacing x^2 + 1 by u, the integral becomes:
∫[(1/2)*sin(u)] du
- Now the integral can be evaluated using trigonometric properties.
Integration by Substitution
- Integration by substitution is a technique used to evaluate indefinite integrals.
- It involves using a suitable substitution to simplify the integrand.
- This method is helpful when dealing with integrals that involve compositions of functions or products of functions.
Steps for Integration by Substitution
- Identify a suitable substitution, typically denoted by u.
- Calculate the derivative of u and substitute it in the integrand.
- Replace all occurrences of the original variable with u.
- Evaluate the resulting integral with respect to u.
- Finally, replace u with the original variable to obtain the solution to the indefinite integral.
Example 1
Evaluate the integral:
∫[cos(x)*sin^2(x)] dx
- Let u = sin(x)
- Differentiating u with respect to x, we get du/dx = cos(x)
- Replacing cos(x) by du/dx, the integral becomes:
∫[u^2] du
- This can be easily integrated to get the final solution.
Example 2
Evaluate the integral:
∫[e^x*cos(e^x)] dx
- Let u = e^x
- Differentiating u with respect to x, we get du/dx = e^x
- Replacing e^x by u, the integral becomes:
∫[cos(u)] du
- Integrating cos(u) with respect to u gives us the solution.
The Chain Rule in Integration
- The chain rule is a fundamental concept in calculus.
- It states that the derivative of a composition of functions can be expressed as the product of the derivative of the outer function and the derivative of the inner function.
- Similarly, the chain rule can be applied to integration, known as the integration by substitution.
Key Steps for Integration by Substitution (Chain Rule)
- Identify a suitable substitution, typically denoted by u.
- Calculate the derivative of u with respect to x (du/dx).
- Express dx in terms of du using the chain rule:
dx = du/(du/dx)
- Replace dx and all occurrences of the original variable with the suitable substitution.
- Integrate the resulting expression with respect to u.
- Finally, replace u with the original variable to obtain the solution to the indefinite integral.
Definite Integral - Introduction to the Method of Substitution
- The method of substitution is a powerful technique used to evaluate definite integrals.
- It simplifies the integrand by making a suitable substitution.
- This method is effective for integrals involving complex algebraic or trigonometric expressions.
Steps for the Method of Substitution
- Identify a suitable substitution variable, often denoted as u.
- Calculate the derivative of u with respect to the original variable.
- Replace the derivative in the integrand expression.
- Substitute all occurrences of the original variable with u.
- Evaluate the new integral with respect to u.
- Finally, substitute back the original variable to obtain the solution to the original integral.
Example 1
Evaluate the integral:
∫[(x^3 + 2x^2 - 1)/(x^2 + x + 1)] dx
- Let u = x^2 + x + 1
- Find the derivative of u: du/dx = 2x + 1
- Replace the expression in the integral with u: ∫[(x^3 + 2x^2 - 1)/u] du
- This integral can now be evaluated using standard integration techniques.
Example 2
Evaluate the integral:
∫[x*sin(x^2 + 1)] dx
- Let u = x^2 + 1
- Find the derivative of u: du/dx = 2x
- Replace the expression in the integral with u: ∫[(1/2)*sin(u)] du
- This integral can now be evaluated using trigonometric properties.
Integration by Substitution
- Integration by substitution is a method used to evaluate indefinite integrals.
- It simplifies the integrand by employing a suitable substitution.
- This method is particularly useful for integrals involving compositions or product of functions.
Steps for Integration by Substitution
- Identify a suitable substitution variable, typically denoted as u.
- Calculate the derivative of u with respect to the original variable.
- Replace the derivative in the integrand expression.
- Substitute all occurrences of the original variable with u.
- Evaluate the new integral with respect to u.
- Finally, substitute back the original variable to obtain the solution to the indefinite integral.
Example 1
Evaluate the integral:
∫[cos(x)*sin^2(x)] dx
- Let u = sin(x)
- Find the derivative of u: du/dx = cos(x)
- Replace the expression in the integral with u: ∫[u^2] du
- This integral can now be solved conveniently.
Example 2
Evaluate the integral:
∫[e^x*cos(e^x)] dx
- Let u = e^x
- Find the derivative of u: du/dx = e^x
- Replace the expression in the integral with u: ∫[cos(u)] du
- This integral can now be evaluated using standard integration techniques.
The Chain Rule in Integration
- The chain rule is a fundamental concept in calculus.
- It states that the derivative of a composition of functions can be expressed as the product of the derivative of the outer function and the derivative of the inner function.
- In integration, the chain rule is applied through integration by substitution.
Key Steps for Integration by Substitution (Chain Rule)
- Identify a suitable substitution variable, often denoted as u.
- Calculate the derivative of u with respect to x (du/dx).
- Express dx in terms of du using the chain rule: dx = du/(du/dx).
- Substitute dx and all occurrences of the original variable with the suitable substitution.
- Integrate the resulting expression with respect to u.
- Finally, substitute u back with the original variable to obtain the solution to the indefinite integral.
Definite Integral - Introduction to the Method of Substitution
- The method of substitution is a technique used to evaluate definite integrals.
- It involves making a suitable substitution to simplify the integrand.
- This method is particularly useful when dealing with integrals that involve complicated algebraic or trigonometric expressions.
Steps for the Method of Substitution
- Identify a suitable substitution variable, typically denoted by u.
- Calculate the derivative of u with respect to x (du/dx).
- Replace the derivative in the integrand expression.
- Substitute all occurrences of the original variable with u.
- Evaluate the new integral with respect to u.
- Finally, substitute back the original variable to obtain the solution to the original integral.
Example
Evaluate the integral:
∫[(x^3 + 2x^2 - 1)/(x^2 + x + 1)] dx
- Let u = x^2 + x + 1.
- Find the derivative of u with respect to x: du/dx = 2x + 1.
- Replace the expression in the integral with u: ∫[(x^3 + 2x^2 - 1)/u] du.
- This integral can now be evaluated using standard integration techniques.
Integration by Substitution
- Integration by substitution is a method used to evaluate indefinite integrals.
- It simplifies the integrand by employing a suitable substitution.
- This method is particularly useful for integrals involving compositions or products of functions.
Steps for Integration by Substitution
- Identify a suitable substitution variable, typically denoted as u.
- Calculate the derivative of u with respect to the original variable.
- Replace the derivative in the integrand expression.
- Substitute all occurrences of the original variable with u.
- Evaluate the new integral with respect to u.
- Finally, substitute back the original variable to obtain the solution to the indefinite integral.
Example
Evaluate the integral:
∫[cos(x)*sin^2(x)] dx
- Let u = sin(x).
- Find the derivative of u with respect to x: du/dx = cos(x).
- Replace the expression in the integral with u: ∫[u^2] du.
- This integral can now be solved using standard integration techniques.
The Chain Rule in Integration
- The chain rule is a fundamental concept in calculus.
- It states that the derivative of a composition of functions can be expressed as the product of the derivative of the outer function and the derivative of the inner function.
- In integration, the chain rule is applied through integration by substitution.
Key Steps for Integration by Substitution (Chain Rule)
- Identify a suitable substitution variable, typically denoted as u.
- Calculate the derivative of u with respect to x (du/dx).
- Express dx in terms of du using the chain rule: dx = du/(du/dx).
- Substitute dx and all occurrences of the original variable with the suitable substitution.
- Integrate the resulting expression with respect to u.
- Finally, substitute u back with the original variable to obtain the solution to the indefinite integral.
Example
Evaluate the integral:
∫[e^x*cos(e^x)] dx
- Let u = e^x.
- Find the derivative of u with respect to x: du/dx = e^x.
- Replace the expression in the integral with u: ∫[cos(u)] du.
- This integral can now be evaluated using standard integration techniques.
Summary
- The method of substitution is used to evaluate definite and indefinite integrals.
- For definite integrals, the steps include identifying a suitable substitution, replacing the variable and evaluating the integral.
- For indefinite integrals, the steps include identifying a suitable substitution, replacing the variable, integrating with respect to the substitution, and substituting back the original variable.
- Integration by substitution uses the chain rule to simplify the integral.
- By following these methods, we can effectively evaluate a variety of integrals.