Calculus

Introduction to integrals

Area as a limit of sums, antiderivatives, and the theorem that connects them.

About 12 minutes

Integration answers two questions that turn out to be the same question. How do you add up infinitely many infinitely small pieces — for instance to find the area under a curve? And which function has a given function as its derivative?

The definite integral as area

Split the interval [a, b] into $n$ strips of width $\Delta x$ , approximate the region under the curve by rectangles, and add their areas. As the strips get thinner the sum approaches a limit: the definite integral. Area below the x-axis counts as negative.

\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i)\,\Delta x

Antiderivatives

An antiderivative of $f$ is a function $F$ whose derivative is $f$ . Antiderivatives are never unique: adding any constant gives another one, because constants differentiate to zero. That is the reason for the “+ C”.

Reversing the power rule gives the power rule for integrals, valid for every exponent except $-1$ , where the answer is a logarithm: \int \frac{1}{x}\,dx = \ln|x| + C.

\int x^{n}\,dx = \frac{x^{n+1}}{n + 1} + C,\quad n \ne -1

The Fundamental Theorem of Calculus

The theorem says that the two ideas coincide: to evaluate a definite integral of a continuous function, find any antiderivative and subtract its values at the two limits. No limit of sums is required.

Not every function has an antiderivative that can be written with elementary functions — e^{-x^2} is a famous example. Such integrals are evaluated numerically, for instance with Simpson’s rule.

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

Worked examples

Find \int 2x\,dx.

Raise the power by one and divide by the new power: 2 \cdot \frac{x^2}{2}.
Simplify and add the constant.

Answer: x^2 + C

Evaluate \int_0^3 x^2\,dx.

An antiderivative is F(x) = \frac{x^3}{3}.
F(3) - F(0) = \frac{27}{3} - 0.

Answer: 9

Common mistakes

Leaving out the constant of integration in an indefinite integral.
Using the power rule for \int \frac{1}{x}\,dx; the result is \ln|x| + C.
Integrating a product factor by factor — there is no simple product rule for integrals.
Applying the Fundamental Theorem across a point where the function is undefined, such as \int_{-1}^{1} \frac{1}{x^2}\,dx.

Practice problems

1.\int (3x^2 + 1)\,dx
2.\int \cos x\,dx
3.\int_0^{\pi} \sin x\,dx
4.\int_1^{e} \frac{1}{x}\,dx

Tools for this lesson

Integral CalculatorIndefinite integrals (antiderivatives) found symbolically and verified by differentiation before they are shown.Definite Integral CalculatorEvaluate definite and improper integrals: analytically when an antiderivative exists, otherwise by adaptive quadrature.Numerical Integration CalculatorApproximate integrals with the trapezoidal, midpoint and Simpson’s rules and compare them with an adaptive reference value.Derivative CalculatorDifferentiate functions symbolically — first, second and higher derivatives — with the rules used and a numerical verification.

Next lesson: Order of operations

The definite integral as area

Split the interval

[a, b]

[a, b] into

n

strips of width

\Delta x

, approximate the region under the curve by rectangles, and add their areas. As the strips get thinner the sum approaches a limit: the definite integral. Area below the x-axis counts as negative.

\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i)\,\Delta x

\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i)\,\Delta x

Antiderivatives

An antiderivative of

f

is a function

F

whose derivative is

f

. Antiderivatives are never unique: adding any constant gives another one, because constants differentiate to zero. That is the reason for the “

+ C

+ C”.

Reversing the power rule gives the power rule for integrals, valid for every exponent except

-1

, where the answer is a logarithm:

\int \frac{1}{x}\,dx = \ln|x| + C

\int \frac{1}{x}\,dx = \ln|x| + C.

\int x^{n}\,dx = \frac{x^{n+1}}{n + 1} + C,\quad n \ne -1

\int x^{n}\,dx = \frac{x^{n+1}}{n + 1} + C,\quad n \ne -1

The Fundamental Theorem of Calculus

Not every function has an antiderivative that can be written with elementary functions —

e^{-x^2}

e^{-x^2} is a famous example. Such integrals are evaluated numerically, for instance with Simpson’s rule.

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

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

Worked examples

Find $\int 2x\,dx$ \int 2x\,dx.

Raise the power by one and divide by the new power: $2 \cdot \frac{x^2}{2}$ 2 \cdot \frac{x^2}{2}.
Simplify and add the constant.

Answer: $x^2 + C$ x^2 + C

Evaluate $\int_0^3 x^2\,dx$ \int_0^3 x^2\,dx.

An antiderivative is $F(x) = \frac{x^3}{3}$ F(x) = \frac{x^3}{3}.
$F(3) - F(0) = \frac{27}{3} - 0$ F(3) - F(0) = \frac{27}{3} - 0.

Answer: 9

Common mistakes

Leaving out the constant of integration in an indefinite integral.

Using the power rule for

\int \frac{1}{x}\,dx

\int \frac{1}{x}\,dx; the result is

\ln|x| + C

\ln|x| + C.

Integrating a product factor by factor — there is no simple product rule for integrals.

Applying the Fundamental Theorem across a point where the function is undefined, such as

\int_{-1}^{1} \frac{1}{x^2}\,dx

\int_{-1}^{1} \frac{1}{x^2}\,dx.

Practice problems

\int (3x^2 + 1)\,dx

\int (3x^2 + 1)\,dx

\int \cos x\,dx

\int \cos x\,dx

\int_0^{\pi} \sin x\,dx

\int_0^{\pi} \sin x\,dx

\int_1^{e} \frac{1}{x}\,dx

\int_1^{e} \frac{1}{x}\,dx