Integrals: complete lesson (definite, antiderivatives, area, properties)

Integral: intuitive definition

For $f$ continuous and positive on $[a, b]$, the integral $\int_a^b f(x)\, dx$ is the area between the graph of $f$, the x-axis, and the lines $x = a$ and $x = b$.

For $f$ with varying sign, areas above the axis count positively and below negatively.

Fundamental theorem

If $F$ is an antiderivative of $f$ on $[a, b]$:

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

The formula that turns an area computation into an antiderivative computation.

Key properties

Linearity: $\int_a^b (\alpha f + \beta g)(x)\, dx = \alpha \int_a^b f(x)\, dx + \beta \int_a^b g(x)\, dx$

Chasles’ relation (additivity): $\int_a^c f(x)\, dx = \int_a^b f(x)\, dx + \int_b^c f(x)\, dx$

Swapping bounds: $\int_a^b f(x)\, dx = - \int_b^a f(x)\, dx$

Positivity: If $f \geq 0$ on $[a, b]$ with $a < b$, then $\int_a^b f(x)\, dx \geq 0$.

Order: If $f \leq g$ on $[a, b]$ with $a < b$, then $\int_a^b f \leq \int_a^b g$.

Mean value

The average value of $f$ on $[a, b]$:

$\bar{f} = \dfrac{1}{b - a} \int_a^b f(x)\, dx$

The “constant height” that would enclose the same area.

Area between two curves

If $f \geq g$ on $[a, b]$:

$\mathcal{A} = \int_a^b \big(f(x) - g(x)\big) \, dx$

Integration by parts (IBP)

$\int_a^b u'(x) v(x)\, dx = \big[ u(x) v(x) \big]_a^b - \int_a^b u(x) v'(x)\, dx$

Useful for $\int x \ln x$, $\int x e^x$, $\int \ln x$, etc.

Common antiderivatives

$f(x)$	$F(x)$
$x^n$ ($n \neq -1$)	$\dfrac{x^{n+1}}{n+1}$
$\dfrac{1}{x}$	$\ln
$e^x$	$e^x$
$\cos x$	$\sin x$
$\sin x$	$-\cos x$

+ constant (for the antiderivative, not for the definite integral).

Useful special cases

• $\int_a^b f = 0$ if $f$ odd and bounds symmetric $[-a, a]$.
• $\int_{-a}^{a} f = 2 \int_0^a f$ if $f$ even.
• $\int_a^a f = 0$ always.

Common mistakes

1. Adding $+C$ to a definite integral (it’s a number).
2. Forgetting to check continuity before using the fundamental theorem.
3. Sign error when swapping bounds.
4. Using $\ln x$ instead of $\ln |x|$ as antiderivative of $1/x$.