Evaluate The Integral Using Substitution Calculator

How to Use This Calculator

To evaluate an integral using substitution:

1. Enter the integrand (the function you want to integrate) in the first field.
2. Specify the substitution variable (u) in the second field.
3. Enter the substitution expression (du/dx) in the third field.
4. Enter the limits of integration (a and b) if evaluating a definite integral.
5. Click "Calculate" to see the result and step-by-step solution.

The calculator will show you the substitution steps, the antiderivative, and the final evaluated result.

The Substitution Method

The substitution method (also called u-substitution or integration by substitution) is a technique for evaluating integrals. It's particularly useful when the integrand contains a function and its derivative.

To use substitution:

1. Identify a substitution u = g(x).
2. Find du/dx and express du in terms of dx.
3. Rewrite the integral in terms of u.
4. Integrate with respect to u.
5. Substitute back in terms of x.

This method is often used with trigonometric, exponential, and logarithmic functions.

Worked Example

Let's evaluate the integral ∫x²e^(x³)dx using substitution.

1. Let u = x³. Then du/dx = 3x², so du = 3x²dx.
2. We can write x²dx = (1/3)du.
3. Substitute: ∫x²e^(x³)dx = ∫e^u(1/3)du = (1/3)∫e^udu.
4. Integrate: (1/3)e^u + C.
5. Substitute back: (1/3)e^(x³) + C.

Common Pitfalls

When using substitution, be careful about these common mistakes:

• Incorrect substitution: Choose u carefully. It should be a function whose derivative appears in the integrand.
• Forgetting to change variables: After integrating with respect to u, remember to substitute back to x.
• Sign errors: When dealing with definite integrals, pay attention to the limits of integration after substitution.
• Missing constants: Don't forget to include the constant of integration (C) when evaluating indefinite integrals.