Integral Calculator with Steps
Type a function and Calvo integrates it step by step, naming the method it uses. Add limits for a definite integral with the exact value and a decimal.
Integrate step by step
Type * for multiplication (2x and x sin x also work), ^ for powers, sqrt(), ln() for natural log, e^x or exp(x). For limits you can use numbers, pi and e.
Methods Calvo uses
| Method | When it is used |
|---|---|
| Power rule | ∫ xn dx = xn+1/(n+1) + C, for n ≠ −1 |
| Sum and constant multiple | Split the integral into terms and pull out constants |
| Standard integrals | ex, 1/x, sin, cos, sec², 1/(x²+a²), with linear arguments such as sin(3x) |
| Substitution (u-sub) | Integrals like ∫ 2x/(x²+1) dx or ∫ x ex² dx |
| Integration by parts | ∫ u dv = uv − ∫ v du, for x·ex, x·sin x, ln x |
| Partial fractions | Rational functions such as 1/((x−1)(x+2)) |
Worked example: integration by parts
Find ∫ x ex dx. Choose u = x and dv = ex dx, so du = dx and v = ex. Then ∫ x ex dx = x ex − ∫ ex dx = x ex − ex + C.
Why there is a “+ C”
An indefinite integral is a whole family of functions that differ by a constant, because the derivative of a constant is zero. Calvo shows the antiderivative and adds + C. For a definite integral (with limits) the constant cancels, so you get a number.
How Calvo checks the answer
After finding an antiderivative, Calvo differentiates it numerically and compares it with your function at several points. For definite integrals it also compares the exact value with a numerical (Simpson) estimate and warns you if they differ.
Limits
Some integrals have no closed form or need tricks Calvo does not use yet, for example ∫ x³√(x²+1) dx or improper integrals. In that case Calvo says so instead of guessing.
FAQ
How do I integrate step by step?
Look for the pattern: a power, a standard function, a product that suits parts, or an inside function whose derivative also appears (substitution). Calvo names the method it uses and shows each line.
What is the integral of x e^x?
x e^x − e^x + C, by integration by parts with u = x and dv = e^x dx.
What is the integral of 1/x?
ln|x| + C. The absolute value keeps the answer valid for negative x as well.
How do I do a definite integral?
Type the lower and upper limits. Calvo finds the antiderivative F and gives F(upper) − F(lower), with the exact value and a decimal.
Why does Calvo say it cannot solve my integral?
Some integrals need a method it does not use yet, or have no elementary antiderivative. It prefers to say so rather than show a wrong answer.
Why is the constant C missing in a definite integral?
In a definite integral F(b) − F(a) cancels the constant, so the result is just a number.
Is my answer allowed to look different from the book?
Yes. Two antiderivatives can differ by a constant, or be written in an equivalent form. Differentiate both to confirm they match.
Does it work offline?
Yes. The calculation runs in your browser and the page is cached after your first visit.