Previously we said that the second part of the Fundamental Theorem of Calculus provides a very powerful method for evaluating the definite integral of a function, assuming that we can find an antiderivative of the function. In this section we introduce a notation for antiderivatives, review the formulas for antiderivatives, and use them to evaluate definite integrals. We also reformulate FTC2 in a way that makes it easier to apply to science and engineering problems.
Both parts of the Fundamental Theorem establish connections between antiderivatives and definite integrals. Part 1 says that if $f$ is continuous, then $\int_a^x f(t)\,dt$ is an antiderivative of $f$. Part 2 says that $\int_a^b f(x)\,dx$ can be found by evaluating $F(b) − F(a)$, where $F$ is an antiderivative of $f$.
We need a convenient notation for antiderivatives that makes them easy to work with. Because of the relation between antiderivatives and integrals given by the Fundamental Theorem, the notation $\int f(x)\,dx$ is traditionally used for an antiderivative of f and is called an indefinite integral. Thus
$$ \int f(x)\,dx = F(x) \qquad \text{means} \qquad F'(x) = f(x) $$
For example, we can write
$$ \int x^2\,dx = \frac{x^3}{3} + C \qquad \text{because} \qquad \frac{d}{dx}\left(\frac{x^3}{3} + C\right) = x^2 $$
So we can regard an indefinite integral as representing an entire family of functions (one antiderivative for each value of the constant $C$).
You should distinguish carefully between definite and indefinite integrals. A definite integral $\int_a^b f(x)\,dx$ is a number, whereas an indefinite integral $\int f(x)\,dx$ is a function (or family of functions).
The connection between them is given by Part 2 of the Fundamental Theorem: if $f$ is continuous on $[a, b]$, then
$$ \int_a^b f(x)\,dx = \int f(x)\,dx \Big]_a^b $$
The effectiveness of the Fundamental Theorem depends on having a supply of antiderivatives of functions. We therefore restate the Table of Antidifferentiation Formulas from Section 4.9, together with a few others, in the notation of indefinite integrals. Any formula can be verified by differentiating the function on the right side and obtaining the integrand. For instance,
$$ \int \sec^2 x\,dx = \tan x + C \qquad \text{because} \qquad \frac{d}{dx}(\tan x + C) = \sec^2 x $$
Indefinite Integrals
$$ \int cf(x)\,dx = c\int f(x)\,dx $$
$$ \int k\,dx = kx + C $$
$$ \int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx $$
$$ \int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) $$
$$ \int e^x\,dx = e^x + C $$
$$
\int \sin x\,dx = -\cos x + C $$
$$ \qquad \int \csc^2 x\,dx = -\cot x + C $$
$$ \int \sec x \tan x\,dx = \sec x + C $$
$$ \int \csc x \cot x\,dx = -\csc x + C $$
$$ \int \frac{1}{x^2+1}\,dx = \tan^{-1}x + C $$
$$ \qquad \int \frac{1}{x}\,dx = \ln|x| + C $$
$$ \qquad \int \cos x\,dx = \sin x + C $$
$$ \int \sec^2 x\,dx = \tan x + C $$
$$ \int \csc x \cot x\,dx = -\csc x +C $$
$$ \int \frac{1}{\sqrt{1-x^2}}\,dx = \sin^{-1}x + C $$
Recall that the most general antiderivative on a given interval is obtained by adding a constant to a particular antiderivative. We adopt the convention that when a formula for a general indefinite integral is given, it is valid only on an interval. Thus we write