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📘 About the Integral Calculator
Integration reverses differentiation: the antiderivative F(x) is a function whose derivative is f(x). Integrals give areas under curves, totals from rates, volumes, work done and probabilities.
This calculator integrates the forms met in school and first-year university: polynomials, powers and roots, eˣ and aˣ, ln x, sin, cos, tan, sec², 1/(1 + x²) and 1/√(1 − x²), rational functions with linear or quadratic denominators (partial fractions), u-substitution and integration by parts. Every answer is differentiated back to make sure it is right.
🧮 Formula
✏️ Worked examples
By parts: x eˣ
You get| Antiderivative | ∫ xe^x dx = xe^x − e^x + C |
| Integrand | xe^x |
| Check: differentiate the answer | d/dx(…) = xe^x |
∫ xe^x dx = xe^x − e^x + C.
Substitution: x³ cos(x⁴)
You get| Antiderivative | ∫ x³·cos(x⁴) dx = sin(x⁴)/4 + C |
| Integrand | x³·cos(x⁴) |
| Check: differentiate the answer | d/dx(…) = x³·cos(x⁴) |
∫ x³·cos(x⁴) dx = sin(x⁴)/4 + C.
🧭 How to use this calculator
What you enter
- Integrate f(x)Use ^ for powers, * or nothing for ×: 3x^2 + 2sin(x), e^(2x), sqrt(x+1), ln(x).
- VariableOne letter. Other letters are treated as constants.
What you get
- Antiderivative
- Integrand
- Check: differentiate the answer
❓ Frequently asked questions
What is the integral of 1/x?
∫1/x dx = ln|x| + C. The absolute value makes it valid for negative x as well.
When do I use integration by parts?
For a product where one factor gets simpler when differentiated (like x, x² or ln x) and the other is easy to integrate (eˣ, sin x, cos x): ∫x eˣ dx = x eˣ − eˣ + C.
Why add + C?
Every function has infinitely many antiderivatives that differ by a constant, because the derivative of a constant is 0.
Which integrals have no answer?
Some, like ∫e^(x²) dx and ∫sin(x)/x dx, cannot be written with ordinary functions. You can still find their value between two limits with the definite integral calculator.
