What is the difference between an indefinite integral and the Fundamental Theorem?

An indefinite integral, \int f(x) \, dx = F(x) + C, finds a general family of antiderivatives. The Fundamental Theorem uses a specific antiderivative and two bounds to compute a definite integral (a single number): \int_a^b f(x) \, dx = F(b) - F(a). In other words, you use indefinite integrals as a tool within the Fundamental Theorem to find the antiderivative you need.

Why do I sometimes get a negative answer, and is that correct?

A negative result means the function dips below the x-axis, so the signed area below the axis outweighs the signed area above. This is the correct behavior of the definite integral: it measures signed area. If your arithmetic is correct, a negative answer is a valid and meaningful result.

Can the Fundamental Theorem always be used to evaluate a definite integral?

The Fundamental Theorem applies when f is continuous on [a, b] and you can find a closed-form antiderivative. Most basic functions (polynomials, exponentials, logarithms, trigonometric functions) have antiderivatives. However, some functions like \frac{\sin(x)}{x} do not have a simple closed-form antiderivative, so you must use numerical approximation instead.

What should I do if I cannot find an antiderivative?

Check whether the integrand matches a standard antiderivative formula or can be rewritten to match one. If it involves a product or composition, try techniques like u-substitution or integration by parts. If an antiderivative still eludes you, approximate the definite integral using a numerical method such as Riemann sums or the Trapezoidal Rule.

Fundamental Theorem of Calculus

Compute the exact value of a definite integral using the antiderivative—connect area under a curve to the accumulation of change.

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

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What each symbol means

What Fundamental Theorem of Calculus takes

Symbol	Meaning
$f$	The original function being integrated; represents a rate of change or the height of a curve at each point. If you apply the Fundamental Theorem formula directly to $f$ instead of its antiderivative, your answer will be wrong.
$F$	The antiderivative of $f$ , also called a primitive; a function whose derivative equals $f$ . If you forget to find $F$ before applying the Fundamental Theorem, you have no way to compute the definite integral.
$a$	The lower limit of integration, the starting point where accumulation begins. If you accidentally swap $a$ with $b$ , your final answer will have the opposite sign.
$b$	The upper limit of integration, the ending point where accumulation stops. If you accidentally swap $b$ with $a$ , your final answer will change sign.
$x$	The variable of integration, the independent variable that $f$ depends on. If you treat $x$ as a constant rather than a variable, you cannot perform the integration.

When to use it

Use the Fundamental Theorem when you need to evaluate a definite integral and you can find an antiderivative of the integrand.

Level

Usually taught in: Calculus I · Appears on: AP Calculus

Worked examples

1. Evaluate a simple integral with small integer bounds

Problem

Evaluate

\int_1^3 2x \, dx

$F(x) = x^2$
Find the antiderivative of $2x$ using the power rule: since the derivative of $x^2$ is $2x$ , we have $F(x) = x^2$ .
$[x^2]_1^3$
Apply the Fundamental Theorem by setting up the bracket notation showing the antiderivative and bounds.
$= 3^2 - 1^2$
Substitute the upper bound and lower bound: compute $F(3) - F(1)$ .
$= 9 - 1 = 8$
Evaluate the arithmetic: $3^2 = 9$ and $1^2 = 1$ , then subtract to get the final answer.

Answer: $8$

This problem demonstrates the core Fundamental Theorem procedure: find the antiderivative, set up brackets with the bounds, substitute the bounds, and subtract. The positive answer reflects that the function $2x$ is positive throughout this interval.

2. Integrate a polynomial with a negative bound and fractional simplification

Problem

Evaluate

\int_{-1}^2 (x^2 - 3x) \, dx

$F(x) = \frac{x^3}{3} - \frac{3x^2}{2}$
Integrate term by term using the power rule: $\int x^2 \, dx = \frac{x^3}{3}$ and $\int 3x \, dx = \frac{3x^2}{2}$ .
$\left[\frac{x^3}{3} - \frac{3x^2}{2}\right]_{-1}^2$
Set up the Fundamental Theorem bracket notation with bounds $-1$ and $2$ .
$F(2) = \frac{2^3}{3} - \frac{3(2)^2}{2} = \frac{8}{3} - 6 = -\frac{10}{3}$
Evaluate at the upper bound $x = 2$ : compute $\frac{8}{3} - 6 = \frac{8}{3} - \frac{18}{3} = -\frac{10}{3}$ using a common denominator.
$F(-1) = \frac{(-1)^3}{3} - \frac{3(-1)^2}{2} = -\frac{1}{3} - \frac{3}{2} = -\frac{11}{6}$
Evaluate at the lower bound $x = -1$ : convert $-\frac{1}{3} - \frac{3}{2}$ to a common denominator to get $-\frac{11}{6}$ .
$F(2) - F(-1) = -\frac{10}{3} - \left(-\frac{11}{6}\right) = -\frac{20}{6} + \frac{11}{6} = -\frac{9}{6} = -\frac{3}{2}$
Subtract: convert to sixths, compute $-\frac{20}{6} + \frac{11}{6} = -\frac{9}{6}$ , then simplify to $-\frac{3}{2}$ .

Answer: $-\frac{3}{2}$

This problem requires careful handling of negative bounds and fraction arithmetic. The negative answer is correct and tells us that on balance, the region where $x^2 - 3x$ is negative (below the x-axis) has greater area than where it is positive. Tracking common denominators throughout is essential.

3. Apply the Fundamental Theorem to a real-world rate problem: phone data consumption

Problem

A phone company charges a monthly base fee of

\$35

plus

\$2

per gigabyte of data used. On day

x

of a 30-day billing cycle, the rate of data consumption is

r(x) = 0.1x

gigabytes per day. How much will the data charges be for the month (not counting the base fee)?

$\text{Total data} = \int_0^{30} 0.1x \, dx$
To find total data consumption over the month, integrate the daily consumption rate from day 0 to day 30.
$F(x) = 0.05x^2$
Find the antiderivative using the power rule: $\int 0.1x \, dx = 0.1 \cdot \frac{x^2}{2} = 0.05x^2$ .
$[0.05x^2]_0^{30} = 0.05(30)^2 - 0.05(0)^2$
Apply the Fundamental Theorem: evaluate the antiderivative at day 30 and day 0.
$= 0.05(900) - 0 = 45 \text{ GB}$
Compute: $0.05 \times 900 = 45$ gigabytes total accumulated over the 30-day period.
$\text{Data cost} = 45 \times 2 = 90 \text{ dollars}$
Multiply total gigabytes by the per-gigabyte cost: $45 \text{ GB} \times \$2/\text{GB} = \$90$ .

Answer: $90 \text{ dollars}$

The Fundamental Theorem transforms a rate function (data per day) into a total (data for the month), which we then use for billing. Since the rate increases throughout the month, integration correctly captures how usage accelerates over time. This shows why FTC is indispensable in applied settings where rates must be accumulated into totals.

Common mistakes

Where Fundamental Theorem of Calculus usually goes wrong

The mistake	Why it is wrong	The fix
Writing $\int_1^3 2x \, dx = 2(3) - 2(1) = 6 - 2 = 4$ .	You applied the Fundamental Theorem formula to the original function $f(x) = 2x$ instead of to its antiderivative $F(x) = x^2$ .	First find the antiderivative: $F(x) = x^2$ , then apply the theorem correctly: $[x^2]_1^3 = 3^2 - 1^2 = 9 - 1 = 8$ .
When computing $\int_0^5 3x^2 \, dx$ , writing $[x^3]_0^5$ and stopping there without evaluating the bounds.	You found the correct antiderivative but failed to complete the Fundamental Theorem procedure by substituting and subtracting at the bounds.	Finish the calculation: $[x^3]_0^5 = 5^3 - 0^3 = 125 - 0 = 125$ .
When evaluating $\int_2^4 x \, dx$ , computing $\left[\frac{x^2}{2}\right]_2^4 = \frac{2^2}{2} - \frac{4^2}{2} = 2 - 8 = -6$ .	You subtracted in reverse order; the Fundamental Theorem requires upper bound minus lower bound, not the other way around.	Compute in the correct order: $\left[\frac{x^2}{2}\right]_2^4 = \frac{4^2}{2} - \frac{2^2}{2} = 8 - 2 = 6$ (a positive answer).

Tips and when to use something else

Always find the antiderivative $F(x)$ first before applying the Fundamental Theorem—never confuse the original function $f$ with its antiderivative.
Remember the order: it is always $F(b) - F(a)$ (upper bound minus lower bound), or your final answer will have the wrong sign.
When integrating a sum or difference of terms, integrate each term separately using the power rule and other standard antiderivative formulas, then combine them before applying the bounds.
If you cannot find an antiderivative by hand or the function is discontinuous on your interval, use numerical methods (Riemann sums or the Trapezoidal Rule) instead of the Fundamental Theorem.

Related formulas

Indefinite IntegralThe indefinite integral finds all functions whose derivative equals a given function, expressed as an antiderivative plus an arbitrary constant.Power Rule for IntegrationPower Rule for Integration instantly finds antiderivatives of polynomial terms; use it to integrate any power of x except x to the negative one.Riemann SumRiemann Sum approximates a definite integral by dividing the area under a curve into rectangles; it connects intuition to formal integration.Definite Integral PropertiesTwo key properties of definite integrals: reversing the bounds negates the integral, and integrating over a single point gives zero.Area Between CurvesFind the area of a region between two curves by computing the definite integral of their difference over a specified interval.

Frequently asked questions

What is the difference between an indefinite integral and the Fundamental Theorem?: An indefinite integral, $\int f(x) \, dx = F(x) + C$ , finds a general family of antiderivatives. The Fundamental Theorem uses a specific antiderivative and two bounds to compute a definite integral (a single number): $\int_a^b f(x) \, dx = F(b) - F(a)$ . In other words, you use indefinite integrals as a tool within the Fundamental Theorem to find the antiderivative you need.
Why do I sometimes get a negative answer, and is that correct?: A negative result means the function dips below the x-axis, so the signed area below the axis outweighs the signed area above. This is the correct behavior of the definite integral: it measures signed area. If your arithmetic is correct, a negative answer is a valid and meaningful result.
Can the Fundamental Theorem always be used to evaluate a definite integral?: The Fundamental Theorem applies when $f$ is continuous on $[a, b]$ and you can find a closed-form antiderivative. Most basic functions (polynomials, exponentials, logarithms, trigonometric functions) have antiderivatives. However, some functions like $\frac{\sin(x)}{x}$ do not have a simple closed-form antiderivative, so you must use numerical approximation instead.
What should I do if I cannot find an antiderivative?: Check whether the integrand matches a standard antiderivative formula or can be rewritten to match one. If it involves a product or composition, try techniques like $u$ -substitution or integration by parts. If an antiderivative still eludes you, approximate the definite integral using a numerical method such as Riemann sums or the Trapezoidal Rule.

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Reviewed 2026-09-18