Definite Integral Evaluator

Math Calculus • Integrals

Written by STEM Calculators Team Published December 28, 2025 Updated February 24, 2026

2. Definite Integral Evaluator

Evaluates \(\int_a^b f(x)\,dx\) using an antiderivative when possible (FTC), with a numerical fallback and a shaded-area plot.

Inputs

Integrand \(f(x)\)

Supported: + − * / ^, parentheses, variable x, constants pi, e, sin cos tan, ln log(base 10), sqrt abs exp. Implicit multiplication is allowed: 2x, (x+1)(x-1), 2sin(x), xsin(x). Trig powers like cos^2(2x) are supported.

Lower bound \(a\)

You can use pi, e, e.g. \(a=0\).

Upper bound \(b\)

Example: \(b=\pi\).

Computation method

If symbolic rules don’t match, use numeric.

Output format

LaTeX is best for readability.

Numeric tolerance

Used in adaptive Simpson.

Graph center \(c\)

Plots use \(x\) on the horizontal axis.

Graph window half-width \(w\)

Plots \(x\in[c-w,c+w]\). Use Auto fit if scales explode.

Constant \(C\) for plotting \(F(x)+C\)

Does not change \(F'(x)\).

Options Show grid Show \(f(x)\) Show \(F(x)+C\) Show \(F'(x)\) Shade area on \([a,b]\) Verify by differentiating (numeric check) Show steps / technique

Presets

Click to auto-fill and compute.

Ready

Graph

Drag to pan • wheel/pinch to zoom • curves: \(f(x)\), \(F(x)+C\), \(F'(x)\). Shaded region shows the signed area on \([a,b]\).

\(f(x)\) \(F(x)+C\) \(F'(x)\)

x: 0, y: 0, zoom(px/unit): 60

Tip: If the plot scales explode, zoom out or use Auto fit.

Result

Enter \(f(x)\), bounds \(a,b\), and click Evaluate.

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2. Definite Integral Evaluator — Theory

What a definite integral means, why it is a signed area, and how the Fundamental Theorem of Calculus (FTC) lets us compute it by antiderivatives.

1) Meaning of \(\int_a^b f(x)\,dx\)

The definite integral represents the accumulated (signed) area between the curve \(y=f(x)\) and the \(x\)-axis from \(x=a\) to \(x=b\). Formally, it is defined as a limit of Riemann sums.

\[ \int_a^b f(x)\,dx = \lim_{n\to\infty}\sum_{k=1}^n f(x_k^*)\,\Delta x, \quad \Delta x=\frac{b-a}{n}. \]

Signed area

If \(f(x)\ge 0\) on \([a,b]\), the integral equals the usual geometric area. If \(f(x)\le 0\) on \([a,b]\), the integral is negative. If the function crosses the axis, the regions below the axis subtract from regions above.

\[ \int_a^b f(x)\,dx = (\text{area above}) - (\text{area below}). \]

Swapping bounds

Reversing the limits changes the sign:

\[ \int_a^b f(x)\,dx = -\int_b^a f(x)\,dx. \]

2) Fundamental Theorem of Calculus (FTC)

If \(F\) is an antiderivative of \(f\) (meaning \(F'(x)=f(x)\)) and \(f\) is continuous on \([a,b]\), then:

\[ \int_a^b f(x)\,dx = F(b)-F(a). \]

Example

Compute \(\int_0^\pi \sin(x)\,dx\).

\[ F(x) = -\cos(x), \quad \int_0^\pi \sin(x)\,dx = F(\pi)-F(0)= -\cos(\pi)-(-\cos(0))= 1-(-1)=2. \]

3) When symbolic FTC may fail

Not every integrand has a simple “closed-form” antiderivative in elementary functions, and rule-based engines can only recognize patterns they were taught (polynomials, basic trig, basic exponentials, etc.). In those cases, numerical integration is used.

Typical symbolic rules (basic)

\[ \int x^n\,dx = \frac{x^{n+1}}{n+1}+C \ (n\ne -1), \quad \int \frac{1}{x}\,dx = \ln|x|+C, \quad \int \sin(x)\,dx = -\cos(x)+C, \quad \int e^{kx}\,dx = \frac{e^{kx}}{k}+C. \]

4) Numerical integration (Simpson’s method)

Simpson’s rule approximates the integral on a small interval by fitting a parabola through sample points. Adaptive Simpson recursively subdivides the interval until the estimated error is below a chosen tolerance.

Simpson on \([a,b]\)

\[ \int_a^b f(x)\,dx \approx \frac{b-a}{6}\left(f(a)+4f\!\left(\frac{a+b}{2}\right)+f(b)\right). \]

Numerical methods can struggle if \(f\) has discontinuities, very steep oscillations, or undefined points inside \([a,b]\). In such cases, the integrand may need to be split into pieces or treated as an improper integral.

5) What the graph shows

The plot shows:

\(f(x)\): the integrand.
\(F(x)+C\): an antiderivative shifted by a constant (this does not affect the definite integral).
\(F'(x)\): the derivative of the plotted antiderivative (should match \(f(x)\) where rules apply).
A shaded region on \([a,b]\) representing the signed area.

Why \(+C\) doesn’t matter

\[ (F(x)+C)'=F'(x), \quad (F(b)+C)-(F(a)+C)=F(b)-F(a). \]

Frequently Asked Questions

What does a definite integral evaluator calculate?

It calculates the accumulated signed area under y = f(x) from x = a to x = b. The result is a single number representing the net area above the x-axis minus the area below it.

How does the Fundamental Theorem of Calculus compute a definite integral?

If an antiderivative F(x) is known for f(x), then the integral from a to b of f(x) dx equals F(b) - F(a). This turns the area problem into evaluating an antiderivative at the bounds.

When will the calculator use numerical integration instead of symbolic FTC?

If the integrand does not match the calculator's symbolic antiderivative rules or has no simple elementary antiderivative, it can fall back to adaptive Simpson integration. Numerical methods approximate the area to within the chosen tolerance.

What does the numeric tolerance control in Simpson integration?

It sets the error target for the adaptive Simpson algorithm. Smaller tolerances generally increase accuracy but may require more subdivisions and computation.

Why does the constant C not change the definite integral value?

Adding a constant shifts an antiderivative up or down, but it cancels in F(b) - F(a). Both endpoints include the same added C, so the difference stays the same.

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