The Fundamental Theorem of Calculus

For example, $x^2$ is an antiderivative of $2x$, since the derivative of $x^2$ is $2x$. The functions $x^2 + 2$ and $x^2 - 2$ are antiderivatives of $2x$ as well. A constant term has derivative $0$, so adding one doesn't change the derivative.

This follows from the Mean Value Theorem: two functions with the same derivative on an interval differ by a constant there. Their graphs are vertical translates of each other, with the same slope at every $x$.

Each basic derivative formula, read backward, gives an antiderivative. Since the derivative of $x^{n+1}$ is $(n + 1)x^n$, dividing by $n + 1$ gives an antiderivative of $x^n$. That works for every $n$ except $n = -1$, where it would divide by zero. Since the derivative of $\cos x$ is $-\sin x$, an antiderivative of $\sin x$ is $-\cos x$.

The row for $\frac{1}{x}$ fills the gap the power rule leaves at $n = -1$. For $x > 0$, the derivative of $\ln x$ is $\frac{1}{x}$. For $x < 0$, $\ln|x| = \ln(-x)$, whose derivative is $\frac{-1}{-x} = \frac{1}{x}$.

To see why, divide $[a, b]$ into $n$ subintervals of width $\Delta x$. The change in $F$ from $a$ to $b$ is the sum of its changes over the subintervals. By the Mean Value Theorem, the change over the $i$-th subinterval is $F'(c_i)\,\Delta x = f(c_i)\,\Delta x$ for some $c_i$ in it. So

for every $n$. The right side is a Riemann sum, and as $n \to \infty$ it approaches the integral of $f$ from $a$ to $b$. The left side doesn't depend on $n$, so it equals that integral.

The difference $F(b) - F(a)$ is written with a bar,

Any antiderivative gives the same difference, since the constant cancels: $\bigl(F(b) + C\bigr) - \bigl(F(a) + C\bigr) = F(b) - F(a)$. So we take $C = 0$.

No row of the table fits an absolute value $|g(x)|$. When $g$ is continuous, split $[a, b]$ where $g$ changes sign. On each piece $|g(x)|$ is $g(x)$ or $-g(x)$, and the theorem applies to each piece.

When $f \ge 0$, the right side is the area of a rectangle on $[a, b]$ of height $f(c)$. In the figure, the part of the region above the rectangle, in darker blue, fills the part of the rectangle above the curve, in amber.

To see why the theorem holds, let $m$ and $M$ be the smallest and largest values of $f$ on $[a, b]$, which exist by the Extreme Value Theorem. Every Riemann sum $\sum f(c_i)\,\Delta x$ lies between $\sum m\,\Delta x = m(b - a)$ and $M(b - a)$, so its limit, the integral, does too. Dividing the integral by $b - a$ gives a number between $m$ and $M$. The values $m$ and $M$ are taken at points of $[a, b]$. The Intermediate Value Theorem on the interval between those two points gives a $c$ where $f$ takes that number.

The height $f(c)$ is a limit of averages of sampled values of $f$. Take $n$ subintervals of width $\Delta x$ and a sample point $c_i$ in each. Since $\frac{1}{n} = \frac{\Delta x}{b - a}$, the average of the $n$ sampled values is

and as $n \to \infty$ the right side approaches the integral divided by $b - a$, which is $f(c)$.

By the Mean Value Theorem for Integrals, $f$ takes its average value at least once on $[a, b]$.