AI Integral Solver
Integration reverses differentiation and measures accumulated change, such as area under a curve. This AI integral solver names the technique at each step, so you can follow how an antiderivative is built.
Indefinite versus definite integrals
An indefinite integral finds a family of functions whose derivative is the one you were given, so it always ends with “+ C”, the constant of integration. A definite integral has limits, such as from 0 to 2, and produces a single number: the signed area between the curve and the x-axis.
The link between them is the Fundamental Theorem of Calculus: find the antiderivative F(x), then compute F(b) − F(a).
Techniques the solver uses
| Technique | When to use it |
|---|---|
| Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C | Polynomials and roots |
| Substitution (u-sub) | A function and its derivative both appear |
| Integration by parts | A product such as x·eˣ or x·sin x |
| Partial fractions | Rational functions with factorable denominators |
Integrals are explained by an AI model, so enter the full expression with dx, for example Integrate x^2 * e^x dx. To check your answer, differentiate it with the calculus solver: you should get the original function back.
Interpreting the result
For a definite integral, a negative value means the area is mostly below the x-axis. To find total area you split at the roots. Visualise the curve first with the graphing solver to see where it crosses zero.
Worked examples
Example 1: Integrate (3x² + 2x) dx
- Apply the power rule to each term: ∫3x² dx = x³ and ∫2x dx = x².
- Add the constant of integration.
Final answer: x³ + x² + C
Example 2: Evaluate the definite integral of x² from 0 to 2
- Antiderivative: x³/3.
- Substitute the limits: 2³/3 − 0³/3 = 8/3.
Final answer: 8/3 ≈ 2.667
Example 3: Integrate x²·eˣ dx (by parts twice)
- Let u = x² and dv = eˣ dx. Then ∫ = x²eˣ − ∫2x·eˣ dx.
- Apply parts again to ∫x·eˣ dx = x·eˣ − eˣ.
- Combine: x²eˣ − 2(x·eˣ − eˣ).
Final answer: eˣ(x² − 2x + 2) + C
Example 4: Integrate 2x·cos(x²) dx (substitution)
- Let u = x², so du = 2x dx.
- The integral becomes ∫cos u du = sin u.
- Replace u with x².
Final answer: sin(x²) + C
Common mistakes to avoid
- Forgetting “+ C” on an indefinite integral.
- Using the power rule on 1/x; its integral is ln|x| + C.
- Not changing the limits (or switching back to x) after a substitution.
- Mixing up u and dv in integration by parts, which makes the integral harder.
- Subtracting the limits in the wrong order, F(a) − F(b).
Frequently asked questions
Why is there a “+ C”?
Many functions share the same derivative, differing by a constant. The C represents any such constant.
How can I check my integral?
Differentiate your answer. If you get the original function, the integral is correct.
Does it solve integrals without an API key?
Integrals need the AI mode. Once an API key is configured in the plugin settings, the solver works through them step by step.
What does a negative definite integral mean?
The net signed area is below the x-axis. The geometric area is the absolute value of each region added together.
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AI-generated solutions are for learning purposes. Always verify important answers.