Integral Calculus Classroom
63.1 The Process of Integration
Integration is fundamentally the reverse process of differentiation. If you have a function $F(x)$ whose derivative is $f(x)$, then $F(x)$ is called an antiderivative or indefinite integral of $f(x)$. Since the derivative of a constant is zero, any constant added to $F(x)$ will still have the same derivative $f(x)$. Thus, we always add a "constant of integration," denoted by $C$, when finding an indefinite integral.
The notation for the indefinite integral of $f(x)$ with respect to $x$ is $\int f(x) dx = F(x) + C$.
63.2 Integrals of $ax^n$
The power rule for integration is a direct reversal of the power rule for differentiation. For any real number $n \neq -1$:
$$ \int x^n dx = \frac{x^{n+1}}{n+1} + C $$
If there is a constant coefficient, it can be pulled out of the integral:
$$ \int ax^n dx = a \int x^n dx = a \frac{x^{n+1}}{n+1} + C $$
For the special case when $n = -1$, i.e., $\int x^{-1} dx = \int \frac{1}{x} dx$, the integral is $\ln|x| + C$.
63.3 Standard Integrals Table
Just like differentiation, there are standard rules for integrating common functions:
- $\int \cos x dx = \sin x + C$
- $\int \sin x dx = -\cos x + C$
- $\int e^x dx = e^x + C$
- $\int a^x dx = \frac{a^x}{\ln a} + C$
- $\int \sec^2 x dx = \tan x + C$
- $\int \csc^2 x dx = -\cot x + C$
- $\int \sec x \tan x dx = \sec x + C$
- $\int \csc x \cot x dx = -\csc x + C$
For more complex functions, substitution or other advanced techniques may be required.
63.4 Definite Integrals
A definite integral $\int_a^b f(x) dx$ represents the net signed area between the function $f(x)$ and the x-axis from $x=a$ to $x=b$. The values $a$ and $b$ are called the lower and upper limits of integration, respectively.
To evaluate a definite integral, we use the Fundamental Theorem of Calculus, which states that if $F(x)$ is an antiderivative of $f(x)$, then:
$$ \int_a^b f(x) dx = F(b) - F(a) $$
Note that the constant of integration $C$ cancels out in definite integrals, so it is usually omitted.