Integrals

Definition

The indefinite integral of $f(x)$ is defined as

$$\int f(x)dx=F(x)+C,\text{when }{F}^{\prime }(x)=f(x),$$

where $C$ is an arbitrary constant.

The definite integral of $f$ over $[a,b]$ is

$${\int }_a^{b}f(x)dx=F(b)-F(a),$$

where $F$ is any antiderivative of $f$, i.e.\ $F^{\prime }(x)=f(x)$. The difference $F(b)-F(a)$ is also written $[F(x){]}_a^b$.

Intuition

The indefinite integral reverses differentiation: it finds a function whose derivative is the given integrand. The constant $C$ reflects the fact that infinitely many antiderivatives exist (shifted vertically), so integration without limits yields a family of functions.

The definite integral measures the signed area between the graph of $f$ and the $x$-axis over $[a,b]$. Riemann sums make this geometric picture precise: partition the interval into thin vertical strips, approximate each strip’s area by a rectangle, and take the limit as the strips become infinitely thin.

Formal Description

Indefinite Integrals

Linearity follows directly from the corresponding differentiation rules:

$$\int af(x)dx=a\int f(x)dx,$$ $$\int [f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx.$$

Combined:

$$\int [a_1{f}_1(x)+\cdots +a_n{f}_n(x)]dx=a_1\int f_1(x)dx+\cdots +a_n\int f_n(x)dx.$$

Standard antiderivatives:

Integrand	Integral
$x^a$ ($a\ne -1$)	$\frac{x^{a+1}}{a+1}+C$
$x^{-1}$	$\ln |x|+C$
$e^{ax}$ ($a\ne 0$)	$\frac{1}{a}{e}^{ax}+C$
$a^x$ ($a>0,a\ne 1$)	$\frac{1}{\ln a}{a}^x+C$

The $x^{-1}$ case uses $\ln (-x)$ for $x<0$ (chain rule gives derivative $x^{-1}$), hence $\ln |x|$ covers both signs.

Differentiating confirms: $\frac{d}{dx}\int f(x)dx=f(x)$.

Definite Integrals

Properties. For $f$ continuous on an interval containing $a$, $b$, $c$:

$$\begin{array}{rl}{\int }_a^{b}f(x)dx& =-{\int }_b^{a}f(x)dx,\\ {\int }_a^{a}f(x)dx& =0,\\ {\int }_a^b\alpha f(x)dx& =\alpha {\int }_a^{b}f(x)dx,\\ {\int }_a^{b}f(x)dx& ={\int }_a^{c}f(x)dx+{\int }_c^{b}f(x)dx.\end{array}$$

Riemann integral. For a bounded function on $[a,b]$, partition $[a,b]$ into $n$ subintervals with widths $\Delta x_i=x_{i+1}-x_i$ and choose ${\xi }_i\in [x_i,x_{i+1}]$. If the Riemann sum

$$\sum _{i=0}^{n-1}f({\xi }_i)\Delta x_i$$

converges as $n\to \infty$ and $\max \Delta x_i\to 0$, then $f$ is Riemann integrable and

$${\int }_a^{b}f(x)dx=\lim \sum _{i=0}^{n-1}f({\xi }_i)\Delta x_i.$$

Every continuous function is Riemann integrable.

Differentiation with respect to the limits. If $F^{\prime }(x)=f(x)$:

$$\frac{d}{dt}{\int }_a^{t}f(x)dx=f(t),\frac{d}{dt}{\int }_t^{b}f(x)dx=-f(t).$$

More generally, for differentiable $a(t)$, $b(t)$ and continuous $f$:

$$\frac{d}{dt}{\int }_{a(t)}^{b(t)}f(x)dx=f(b(t))b^{\prime }(t)-f(a(t))a^{\prime }(t).$$

Applications

• Computing areas, volumes, arc lengths, and surface areas in geometry.
• Finding displacement from velocity (and velocity from acceleration) in physics.
• Evaluating probability densities and cumulative distribution functions.
• Solving differential equations via direct integration.

Trade-offs

• Antiderivatives always exist for continuous functions but need not be expressible in closed form using elementary functions (e.g.\ $\int e^{x^2}dx$, $\int e^{-x^2}dx$, $\int \frac{e^x}{x}dx$).
• The Riemann integral requires boundedness; unbounded functions or infinite intervals require improper integrals with separate convergence analysis.
• The dummy variable $x$ in ${\int }_a^{b}f(x)dx$ carries no meaning outside the integral; confusing it with a free variable is a common error.
• Splitting limits at a point $c$ outside $[a,b]$ is valid but requires $f$ to be integrable on the larger interval.

Links

• Fundamental Theorem of Calculus
• Integration by Substitution
• Integration by Parts