# Integration by Substitution

## 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

- Identify a suitable expression within the integrand to substitute with a new variable, typically denoted as u.
- Differentiate the substituted expression to find the differential du in terms of dx.
- Replace the original expression and differential in the integral with u and du, respectively.
- Evaluate the simplified integral in terms of u.
- 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.

Get your questions answered instantly by an AI Calculus tutor.## 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.