Standard Integration Solver

Integration is a fundamental concept in calculus, often introduced as the reverse process of differentiation. If differentiation is about finding the rate of change (slope), integration is about accumulating quantities or finding the "area under a curve."

If $\frac{d}{dx}(F(x)) = f(x)$, then $F(x)$ is called an antiderivative or indefinite integral of $f(x)$, denoted as:

$$ \int f(x) \, dx = F(x) + C $$

The '$+ C$' represents the constant of integration, because the derivative of a constant is zero. This means there's a whole family of functions that have the same derivative.

Key applications of integration include:

• Calculating areas between curves.
• Finding volumes of solids.
• Determining work done by a variable force.
• Solving differential equations.
• Many applications in physics, engineering, economics, and other sciences.

Basic Forms

• $\int k \, dx = kx + C$ (where $k$ is a constant)
• $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)
• $\int \frac{1}{x} \, dx = \ln|x| + C$
• $\int e^{ax} \, dx = \frac{1}{a}e^{ax} + C$
• $\int a^x \, dx = \frac{a^x}{\ln a} + C$ (for $a > 0, a \neq 1$)

Trigonometric Forms (argument $ax$)

• $\int \sin(ax) \, dx = -\frac{1}{a}\cos(ax) + C$
• $\int \cos(ax) \, dx = \frac{1}{a}\sin(ax) + C$
• $\int \sec^2(ax) \, dx = \frac{1}{a}\tan(ax) + C$
• $\int \csc^2(ax) \, dx = -\frac{1}{a}\cot(ax) + C$
• $\int \sec(ax)\tan(ax) \, dx = \frac{1}{a}\sec(ax) + C$
• $\int \csc(ax)\cot(ax) \, dx = -\frac{1}{a}\csc(ax) + C$

Integrals leading to Logarithms (argument $ax$)

• $\int \tan(ax) \, dx = \frac{1}{a}\ln|\sec(ax)| + C = -\frac{1}{a}\ln|\cos(ax)| + C$
• $\int \cot(ax) \, dx = \frac{1}{a}\ln|\sin(ax)| + C$
• $\int \sec(ax) \, dx = \frac{1}{a}\ln|\sec(ax) + \tan(ax)| + C$
• $\int \csc(ax) \, dx = \frac{1}{a}\ln|\csc(ax) - \cot(ax)| + C$

A definite integral, denoted $\int_a^b f(x) \, dx$, represents the signed area between the curve $y=f(x)$, the x-axis, and the vertical lines $x=a$ and $x=b$.

The Fundamental Theorem of Calculus (Part 2) provides the method for evaluating definite integrals:

If $F(x)$ is an antiderivative of $f(x)$ (i.e., $F'(x) = f(x)$), then:

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

Here, $a$ is the lower limit and $b$ is the upper limit of integration.

Unlike indefinite integrals which result in a family of functions ($F(x)+C$), definite integrals evaluate to a single numerical value.