Reverse power rule
Integrate 3x^2 - 4x
Method: Increase each exponent by one and divide by the new exponent.
- 1Integrate 3x^2 to obtain x^3.
- 2Integrate -4x to obtain -2x^2.
- 3Add the constant of integration.
Final answer
Free online calculator
Integral Calculator
Find antiderivatives and definite integrals with detailed explanations of each integration rule.
Integral mode
Integral
=
Step-by-step solutions
Clear, detailed explanations
Instant results
Accurate answers in seconds
Free to use
No sign-up required
How to Use the Integral Calculator
1
Enter the integrand
Type the expression you want to integrate.
2
Choose the variable
We detect the integration variable automatically.
3
Get the antiderivative
See the final answer and each integration step.
Example
How the Integral Calculator Chooses a Method
Find an antiderivative and learn why a substitution, identity, or integration rule works.
The calculator first looks for linearity and standard antiderivatives, then considers substitution, products, and common trigonometric or exponential patterns. Indefinite integrals include a constant because many functions share the same derivative.
Supported problem types
- Polynomial and power-function antiderivatives
- Common trigonometric, exponential, and logarithmic integrands
- Basic substitution and integration-by-parts patterns
Limits and assumptions
- Some elementary-looking integrals have no elementary antiderivative.
- Always distinguish an indefinite integral from one with numeric bounds.
Worked Integral Calculator Examples
Each example names the method, shows the ordered reasoning, and keeps the final answer separate so you can check your own work.
Substitution
Integrate 2x cos(x^2)
Method: Use u=x^2.
- 1Let u=x^2, so du=2x dx.
- 2Rewrite the integral as the integral of cos(u).
- 3Integrate and substitute x^2 back for u.
Final answer
Integration by parts
Integrate x e^x
Method: Use integration by parts with u=x.
- 1Choose u=x and dv=e^x dx.
- 2Then du=dx and v=e^x.
- 3Apply uv minus the integral of v du.
Final answer
Learn This Topic
What the Integral Calculator Helps You Understand
Find antiderivatives and definite integrals with detailed explanations of each integration rule.
Learning focus 1
Type the expression you want to integrate.
Learning focus 2
We detect the integration variable automatically.
Learning focus 3
See the final answer and each integration step.
Input tips
- • Type the expression you want to integrate.
- • Yes. Indefinite integrals include the constant of integration and explain why it is needed.
Common mistakes to avoid
- • Forgetting the constant of integration for an indefinite integral.
- • Using a substitution without transforming the differential.
Frequently Asked Questions
Does the integral calculator show +C?
Yes. Indefinite integrals include the constant of integration and explain why it is needed.
Can I integrate trigonometric functions?
Yes. sin, cos, tan, and their inverses are all supported.
Why does an indefinite integral need +C?
Differentiation removes constants, so every antiderivative represents a family of functions differing by a constant.
What if the antiderivative is not elementary?
The calculator may return a special-function form or explain that a numerical method or advanced representation is needed.
More Calculators
Derivative CalculatorLimit CalculatorEquation SolverDefinite Integral CalculatorAsymptote Calculator
Browse all calculators