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This definite integral calculator finds the value of \(\int_a^b f(x)\,dx\) and shows how to get it. You will see the antiderivative, the plug-in step \(F(b) – F(a)\), the exact answer (like \(\frac{15}{4}\) or \(\pi\)), a decimal, and a shaded graph of the area.
Every answer is checked twice. First, the antiderivative is checked by differentiating it. Second, the final value is compared with a separate numerical method. If both agree, you see ✓ Verified.
How to Use It
- Type the function, for example
x^2orsin(x). - Enter the lower limit (a) and the upper limit (b). You can use numbers, fractions like
1/2,pi, orinffor infinity. - Press Solve. The answer appears at the top. Open the steps or the graph below it.
What Does a Definite Integral Mean?
A definite integral measures signed area. Area above the x-axis counts as positive. Area below the x-axis counts as negative. So \(\int_0^{2\pi} \sin x\,dx = 0\), because the part above and the part below cancel out.
It can also mean a total amount. If \(v(t)\) is speed, then \(\int_0^5 v(t)\,dt\) is the distance traveled in 5 seconds. See more uses in what calculus is used for in real life.
The Fundamental Theorem of Calculus
This theorem is the shortcut the calculator uses. If \(F\) is any antiderivative of \(f\), then
\[\int_a^b f(x)\,dx = F(b) – F(a).\]
That is why you do not need + C here. The C would appear in both \(F(b)\) and \(F(a)\), and it cancels.
Steps to Evaluate a Definite Integral by Hand
- Find an antiderivative \(F(x)\). Use the antiderivative rules.
- Plug in the upper limit: \(F(b)\).
- Plug in the lower limit: \(F(a)\).
- Subtract: \(F(b) – F(a)\).
Worked Examples
Example 1: \(\int_0^3 x^2\,dx\)
An antiderivative is \(\frac{x^3}{3}\). So \(\frac{3^3}{3} – \frac{0^3}{3} = 9 – 0 = 9\).
Answer: 9.
Example 2: \(\int_0^1 2x(x^2+1)^3\,dx\)
Use u-substitution with \(u = x^2 + 1\), so \(du = 2x\,dx\). The limits change too: when \(x = 0\), \(u = 1\); when \(x = 1\), \(u = 2\).
\(\int_1^2 u^3\,du = \frac{u^4}{4}\Big|_1^2 = 4 – \frac14 = \frac{15}{4}\).
Answer: \(\frac{15}{4} = 3.75\).
Example 3: \(\int_0^1 \frac{1}{1+x^2}\,dx\)
An antiderivative is \(\tan^{-1} x\). So \(\tan^{-1}(1) – \tan^{-1}(0) = \frac{\pi}{4} – 0\).
Answer: \(\frac{\pi}{4} \approx 0.7854\).
Useful Properties
| Property | Formula |
|---|---|
| Same limits | \(\int_a^a f(x)\,dx = 0\) |
| Swap limits | \(\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx\) |
| Split the interval | \(\int_a^c f\,dx = \int_a^b f\,dx + \int_b^c f\,dx\) |
| Constant multiple | \(\int_a^b k f(x)\,dx = k\int_a^b f(x)\,dx\) |
| Odd function, symmetric limits | \(\int_{-a}^{a} f(x)\,dx = 0\) |
What If the Function Blows Up?
Sometimes the function goes to infinity inside the interval, like \(\frac{1}{\sqrt{x}}\) at \(x = 0\). Or one of the limits is infinity. These are improper integrals. This calculator spots them and checks if the answer is finite (converges) or not (diverges). For more practice, use the improper integral calculator.
Common Mistakes
- Subtracting in the wrong order. It is always top limit minus bottom limit: \(F(b) – F(a)\).
- Forgetting to change limits in u-substitution. Either change the limits to u-values, or switch back to x before plugging in.
- Thinking a negative answer is wrong. A negative value just means more area is below the x-axis.
- Ignoring a vertical asymptote. \(\int_{-1}^{1} \frac{1}{x^2}\,dx\) is not \(-2\). It diverges because the function blows up at 0.
Learn the Math Behind the Definite Integral Calculator
A definite integral turns an antiderivative into a number: the signed area between a curve and the x-axis over an interval.
- GuideDefinite vs indefinite integrals, explained
- GuideAntiderivative of sin x and the area under a sine curve
- GuideWhat calculus is used for in real life
- Key termsDefinite integral, Fundamental theorem of calculus, Riemann sum
- Next toolsIntegral Calculator, Improper Integral Calculator, Double Integral Calculator
Frequently Asked Questions
Why is there no + C in a definite integral?
Because it cancels. \((F(b) + C) – (F(a) + C) = F(b) – F(a)\).
Can I use pi or e as a limit?
Yes. Type pi, pi/2, e, or even 2pi.
Why does the calculator show a fraction and a decimal?
The exact value (like \(\frac{\pi}{4}\)) is what most teachers want. The decimal helps you check your work on a regular calculator.
What if there is no antiderivative formula?
Some functions, like \(e^{-x^2}\), have no simple antiderivative. The calculator still gives you an accurate decimal value using numerical integration.
Related Guides
- Definite vs indefinite integrals: a simple comparison
- How to check your antiderivative answer
- Antiderivative practice problems