Definite Integral Calculator

Evaluate an integral between two bounds and separate signed accumulation from geometric area.

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Enter an integral with bounds, such as integrate x from -1 to 2

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Bounds turn an antiderivative into a value

A definite integral measures signed accumulation across an interval. Contributions above the horizontal axis are positive and contributions below it are negative, so the result is not always the total geometric area.

When the integrand is continuous and an antiderivative is available, the Fundamental Theorem of Calculus reduces the problem to F(b) - F(a). Discontinuities and infinite bounds require an improper-integral limit instead of direct substitution.

How to use the definite integral calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

The Fundamental Theorem of Calculus

Find an antiderivative, evaluate it at the upper bound, then subtract its value at the lower bound.

\int_{a}^{b} f (x) d x = F (b) - F (a), F^{'} (x) = f (x)

A reliable way to work through it

Check the interval

Look for discontinuities, infinite bounds, or sign changes before applying an antiderivative mechanically.

Evaluate upper minus lower

Use F(b) - F(a), keeping exact values until the final simplification.

Interpret the sign

State whether the result is signed accumulation or whether the problem instead asks for total geometric area.

Worked example

Evaluate ∫ from -1 to 2 of x dx

F (x) = \frac{x ^{2}}{2}

Find an antiderivative of x.

F (2) - F (- 1) = \frac{2 ^{2}}{2} - \frac{( - 1 ) ^{2}}{2}

Substitute the upper bound, then the lower bound.

2 - \frac{1}{2} = \frac{3}{2}

Simplify the signed value.

The definite integral is 3/2. The total geometric area is 5/2 because the part below the axis must be counted positively for area.

Common mistakes to check

Reversing the subtraction

The Fundamental Theorem uses F(upper) - F(lower). Reversing that order changes the sign.

Calling every result area

A definite integral is signed. Total area requires splitting where the function changes sign.

Ignoring an interior singularity

A vertical asymptote inside the interval can make the integral improper or divergent.

Integral CalculatorFind an indefinite integral, follow the chosen method, and keep the constant of integration.Derivative CalculatorDifferentiate a function, identify each rule, and simplify the result one step at a time.Limit CalculatorStudy what a function approaches from the left, right, or at infinity, even when direct substitution fails.Series Convergence CalculatorChoose a valid convergence test, check its conditions, and classify an infinite series.

Questions students ask

Is a definite integral always area?

It is signed accumulation. Total geometric area requires splitting at sign changes and integrating absolute value.

What happens when the upper bound is smaller?

Reversing the bounds reverses the sign of the definite integral.

Why is there no + C in the answer?

Any antiderivative constant cancels in F(b) - F(a), so the final definite value has no + C.

Can a bound be infinity?

Yes, but that creates an improper integral that must be defined and tested through a limit.