What is Integration by Substitution?

Integration by substitution, also known as u-substitution or change of variables, is a technique used to simplify and evaluate integrals by substituting a complex expression with a simpler variable. This method is particularly useful when the integrand is a composite function or contains expressions that can be transformed into simpler forms.

Common Terms and Definitions

Integrand: The function being integrated, typically denoted as f(x) in the integral ∫f(x)dx.

Composite Function: A function that is formed by applying one function to the result of another function, such as f(g(x)).

Substitution Variable: The new variable, often denoted as u, that replaces a complex expression in the integrand.

Differential: The small change in a variable, denoted as du or dx, that relates the original and substituted variables.

Steps for Integration by Substitution

1. Identify a suitable expression within the integrand to substitute with a new variable, typically denoted as u.
2. Differentiate the substituted expression to find the differential du in terms of dx.
3. Replace the original expression and differential in the integral with u and du, respectively.
4. Evaluate the simplified integral in terms of u.
5. Substitute back the original expression for u to obtain the final result in terms of the original variable.

Examples of Integration by Substitution

Example 1: Evaluate ∫x sin(x²) dx

Let u = x², then du = 2x dx or (du)/2 = x dx

Substituting: ∫x sin(x²) dx = ∫(1/2) sin(u) du

Evaluating: ∫(1/2) sin(u) du = -(1/2) cos(u) + C

Substituting back: -(1/2) cos(x²) + C

Example 2: Evaluate ∫(x + 1)e^(x²+2x) dx

Let u = x² + 2x, then du = (2x + 2) dx or du/2 = (x + 1) dx

Substituting: ∫(x + 1)e^(x²+2x) dx = ∫(1/2)e^u du

Evaluating: ∫(1/2)e^u du = (1/2)e^u + C

Substituting back: (1/2)e^(x²+2x) + C

Common Questions and Answers

How do I choose the right expression to substitute?

Look for expressions that can be simplified by differentiation, such as polynomials, exponentials, or trigonometric functions. The goal is to transform the integrand into a simpler form that is easier to integrate.

What if the integral contains multiple expressions that could be substituted?

Choose the expression that simplifies the integrand the most. In some cases, you may need to perform multiple substitutions to fully simplify the integral.

What if the substitution doesn't seem to work?

Double-check your substitution and differentiation steps. If the substitution doesn't simplify the integrand, try a different expression or consider using other integration techniques, such as integration by parts or partial fractions.

Conclusion

Integration by substitution is a powerful technique for evaluating integrals that involve composite functions or complex expressions. By mastering the steps and practicing with various examples, you'll be well-equipped to tackle a wide range of integration problems in calculus. Remember to choose appropriate substitutions, differentiate correctly, and substitute back to express your final answer in terms of the original variable.