Evaluate Integral Using Substitution Calculator

How to Use This Calculator

Enter the integral you want to evaluate in the form ∫f(x)dx or ∫f(x)dx from a to b. The calculator will:

1. Identify a substitution that simplifies the integral
2. Perform the substitution and simplify
3. Evaluate the integral using the Fundamental Theorem of Calculus
4. Display the result with steps

For definite integrals, the calculator will also show the area under the curve graphically.

The Substitution Method

The substitution method is a technique for evaluating integrals by reversing the chain rule. It's particularly useful when the integrand is a composite function.

When to Use Substitution

Use substitution when the integrand contains a function and its derivative. For example, in ∫x²cos(x³)dx, x² is the derivative of x³.

The Substitution Process

1. Choose u = the inner function (the one whose derivative appears elsewhere in the integrand)
2. Find du = the derivative of u with respect to x
3. Express dx in terms of du: dx = du/dx
4. Rewrite the integral in terms of u
5. Integrate with respect to u
6. Substitute back for u

Common Substitution Patterns

• ∫f(ax + b)dx → u = ax + b
• ∫f(x)/f'(x)dx → u = f(x)
• ∫f(x)√(a² - x²)dx → u = √(a² - x²)

Worked Example

Let's evaluate ∫x²cos(x³)dx using substitution.

Step 1: Choose u

We choose u = x³ because its derivative, 3x², appears in the integrand.

Step 2: Find du

du = 3x²dx → dx = du/3x²

Step 3: Rewrite the integral

∫x²cos(x³)dx = ∫cos(u) du/3x²

Step 4: Integrate

∫cos(u) du = sin(u) + C

Step 5: Substitute back

sin(u) + C = sin(x³) + C

Common Pitfalls

• Forgetting to multiply by dx when finding du
• Choosing u incorrectly (should be a function of x)
• Not substituting back for x after integration
• Missing the +C constant of integration
• Incorrectly handling definite integrals with substitution