Integral calculator (definite and indefinite)
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Result
The antiderivative of 1/x is ln|x|: the absolute value makes it valid for negative x as well.
Find an antiderivative or evaluate a definite integral. For sums of standard terms, such as powers of x, 1/x, e^(ax + b), sin, cos and sec² of ax + b, (ax + b)ⁿ and ln(ax + b), the calculator integrates term by term and evaluates F(b) − F(a) exactly. Any other continuous function is integrated numerically with the adaptive Simpson rule, and the calculator warns about improper integrals.
Step-by-step solution
Read the function and the limits
Find an antiderivative, term by term
Evaluate F(b) − F(a)
Numerical check
Key formulas
- Power rule
- ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C
- Definite integral
- ∫ab f dx = F(b) − F(a)
- Common integrals
- ∫eˣ dx = eˣ, ∫1/x dx = ln|x|, ∫sin x dx = −cos x
- Area between curves
- ∫ab (f − g) dx
How to evaluate a definite integral
- Split the integrand into a sum of terms and integrate each one separately; constant factors stay in front.
- Use the power rule ∫ xⁿ dx = xⁿ⁺¹/(n + 1) for n ≠ −1, and ∫ 1/x dx = ln|x|.
- For a function of ax + b divide by a: ∫ e^(ax + b) dx = e^(ax + b)/a and ∫ cos(ax + b) dx = sin(ax + b)/a.
- Evaluate the antiderivative at the upper and lower limits and subtract: ∫ f(x) dx from a to b = F(b) − F(a). The constant C cancels.
- Check that the integrand is continuous on [a, b]. If it has a vertical asymptote there, as 1/x at x = 0, the integral is improper and needs a limit.
Questions and answers
Which integrals does the calculator solve exactly?
Sums of constant multiples of xⁿ (any rational n), 1/x, e^(ax + b), kˣ, sin(ax + b), cos(ax + b), sec²(ax + b), csc²(ax + b), sec·tan and csc·cot of ax + b, (ax + b)ⁿ and ln(ax + b). Products such as x·eˣ need integration by parts and are evaluated numerically instead.
How accurate is the numerical integration?
The adaptive Simpson rule keeps halving the parts of the interval where the estimate changes, until it agrees to about ten significant digits for smooth functions. The calculator shows the number of function evaluations and an error estimate.
What is the + C?
The constant of integration. Every function has infinitely many antiderivatives that differ by a constant, because the derivative of a constant is zero. In a definite integral it cancels.
Why do I get an error about an improper integral?
The integrand is undefined or infinite somewhere in [a, b], for example 1/x on [−1, 1] or tan(x) across π/2. Such integrals need a limit and may diverge, so the calculator does not give a value for them.
Can I use π in the limits?
Yes. Type pi, pi/2 or 2pi, or e, as a limit; for example the integral of sin(x) from 0 to pi is 2.
Practise it yourself
Practise Integrals
A calculator gives the answer. In the exam you need the method: practise it with new problems, hints and worked solutions.