Integration of formulas with examples

Integration of formulas

Have you ever applied to concept of integration to mathematical formulas? This article will reveal that integration can be used to determine the area under the curve of equations. This area can for example represent total earnings, energy consumption or some other calculated quantity. Integration is therefore an important concept to consider.

What is integration?

Integration is the process of determining the area under the curve of an equation. This is the opposite of differentiation.

The antiderivative (F) is a function whose derivative turns out to be the original function (f).

Standard notation

The standard notation of an integral is formed by a sign that looks like the letter S. The total area (A) under the curve of a graph is then calculated between the lower (a) and upper (b) border of x. Integrals with borders over a specific interval are called definite integrals.

\int_{a}^{b} F(x) dx=A

The antiderivative is calculated by integration of the original function. Also note that no x interval is considered when you determine the antiderivative equation. General solutions of integrals without borders are called indefinite integrals.

\int f(x) dx=F(x)

The original equation is returned if you take the derivative of the antiderivative.

f(x)=[F(x)]'=\frac{dF(x)}{dx}

Area under curves

The total area (A) can be approximated by dividing the graph into small rectangles and adding these up.

A=\sum_{i=1}^{\infty} {\triangle x_i} \cdot {\triangle y_i}

The total sum of all the small rectangles is approximately equal to the definite integral.

A=\int_{a}^{b} F(x) dx

integration of formulas

Basic integration formulas

The integration of basic formulas take the following form. Also note that the constant (C) of the antiderivative (F) would drop out of the equation after differentiation to the original formula (f).

(0)  \int f(x) dx=F(x)

(1)  \int x^n dx=\frac{1}{n+1} x^{n+1} + C

(2)  \int \frac{1}{x} dx=ln(x) + C

(3)  \int a^{x} dx=\frac{a^{x}}{ln(a)} + C

(4)  \int e^{x} dx=e^{x} + C

(5)  \int ln(x) dx=x \cdot ln(x) - x + C

(6)  \int log(x) dx= \frac{x \cdot ln(x) - x}{ln(a)} + C

(7)  \int sin(x) dx = -cos(x) + C

(8)  \int cos(x) dx = sin(x) + C

(9)  \int tan(x) dx = -ln(cos(x)) + C

Example of integrations

X to the power of

(1)  \int x^3 dx = \frac{1}{4} x^{4} + C

Fractions

(2)  \int \frac{1}{x} dx = ln(x) + C

Power equations

(3)  \int 3^{x} dx = \frac{3^{x}}{ln(3)} + C

Exponentials

(4)  \int e^{x} dx = e^{x} + C

Natural logarithmic

(5)  \int ln(x) dx = x \cdot ln(x) - x + C

Logarithmic

(6)  \int log(x) dx = \frac{x \cdot ln(x) - x}{ln(a)} + C

Trigonometric

(7)  \int sin(x) dx = -cos(x) + C

(8)  \int cos(x) dx = sin(x) + C

(9)  \int tan(x) dx = -ln(cos(x)) + C


@ Prerequisites

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