Integration by substitution

Definition

Integration by substitution is a technique used to simplify the process of integrating functions by changing variables. This method often makes it easier to evaluate integrals by transforming them into a more manageable form, especially when dealing with special functions. By using a new variable, the integral can often be expressed in terms of simpler or more familiar functions, leading to a solution that might be difficult to find otherwise.

5 Must Know Facts For Your Next Test

1. Integration by substitution is particularly useful for integrals involving composite functions, where one function is nested inside another.
2. The basic formula used in integration by substitution is: if $u = g(x)$, then $du = g'(x)dx$, allowing you to rewrite the integral in terms of $u$.
3. This technique can also be applied in definite integrals, but it's crucial to change the limits of integration according to the new variable.
4. Common choices for substitution include trigonometric identities or exponential functions, which can simplify integrals significantly.
5. This method not only aids in solving complex integrals but also provides insight into the relationships between different functions involved.

Review Questions

• How does integration by substitution simplify the evaluation of integrals involving composite functions?
  • Integration by substitution simplifies the evaluation of integrals involving composite functions by allowing you to replace complex expressions with a single variable. By choosing an appropriate substitution that captures the structure of the composite function, you can transform the integral into a simpler form that is easier to work with. This method effectively reduces the complexity of the problem, enabling straightforward integration and clearer understanding of the relationships among the involved functions.
• Discuss how you would apply integration by substitution when working with definite integrals and what steps are essential for maintaining accuracy.
  • When applying integration by substitution to definite integrals, it's essential first to make the substitution as usual. However, after substituting the variable, you must also change the limits of integration to correspond with the new variable. This involves calculating the values of the original limits using your substitution function. Once the limits are updated, you can evaluate the integral in terms of the new variable and then substitute back to find the final answer within the original bounds.
• Evaluate how integration by substitution relates to special functions and provide an example illustrating this relationship.
  • Integration by substitution is deeply connected to special functions because many integrals involving these functions can be simplified using this technique. For example, consider the integral of $ ext{sin}^2(x)$; using the identity $ ext{sin}^2(x) = rac{1 - ext{cos}(2x)}{2}$ allows us to simplify the integral. By substituting $u = ext{cos}(2x)$, we transform this into a simpler integral involving $u$. This showcases how substitution not only aids in solving but also highlights interactions between different types of functions.

Related terms

Change of Variables: A mathematical technique that involves substituting one variable for another to simplify calculations or equations.

Definite Integral: An integral that computes the accumulation of quantities over an interval, providing a numerical result that represents the area under a curve between two points.

Special Functions: Functions that have specific properties and applications, such as exponential functions, logarithmic functions, and trigonometric functions, which often arise in integration.