# Indefinite integration

## Anti-Differentials

Anti-differentiation is an operation that does as it sounds: it reverses the differentiation process. Notice, that the respective anti-derivatives must include a constant, since the tangent to a curve refers to a potentially infinite number of curves with that tangent.

One notation method often used is: F(x) is the antiderivative of f(x).

The anti-differential is also referred to as the 'integral', and more usually the symbol ∫, a contribution by Gottfried Leibniz, a co-discoverer of calculus, is used to denote the integral of the function f(x).

If \$F'(x) = f(x)\$ then \$∫f(x)dx = F(x) + c\$

where \$f(x)\$ is the integrand, and \$x\$ is the variable of integration.

\$∫k.f(x)dx = k∫f(x)dx\$

The integral of a sum is the sum of the integrals:

\$\$∫[f(x) + g(x)]dx = ∫f(x)dx + ∫g(x)dx\$\$

## Rules and examples of anti-derivatives

Function f(x) Integral ∫f(x).dx
\$\$f(x) = ax^n\$\$ \$\$F(x) = {ax^(n+1)}/(n+1) + c\$\$
\$\$f(x) = 8x^3\$\$ \$\$F(x) = {8x^(3+1)}/(3+1) + c = 2x^4 + c\$\$
\$\$f(x) = x^½\$\$ \$\$F(x) = {x^(½+1)}/(½+1) + c = {2x^1.5}/3 + c\$\$
\$\$f(x) = e^x\$\$ \$\$F(x) = e^x + c\$\$
\$\$f(x) = 1/x\$\$ \$\$F(x) = lnx + c\$\$
\$\$f(x) = 1/{ax+b}\$\$ \$\$F(x) = 1/aln|ax+b| + c\$\$
\$\$f(x) = e^{2x}\$\$ \$\$F(x) = ½e^{2x} + c\$\$
\$\$f(x) = {1}/(√x) = x^(-½)\$\$ \$\$F(x) = {x^(-½+1)}/(-½+1) + c = 2√x + c\$\$
\$\$f(x) = lnx\$\$ \$\$F(x) = x⋅lnx - x + c\$\$

## Indefinite Integration

### Method of Exhaustion

This famous method of Archimedes to find a value for π brought the Ancient Greek mathematician very close to the invention of calculus.

By drawing ever narrower chords around the inside of a circle, and comparing their collective lengths to a similar series of tangents outside the circle, Archimedes was able to arrive at an estimate of π that lies between two limits.

This fascinating story is told in detail in the book Vitruvian Boy, a novel about the history of mathematics, by Andrew Bone.

The integration of a function produces another function with a constant. This constant transposes the function in the y-axis, producing an infinite number of curves. Each of these curves has the same differential, or function of the slope of the tangent.

Therefore, a specific function may be derived if one set of points which lie on the function are known. In other words, given one value, the value of c may be determined. If the integral function passes through the origin, c = 0.

### Example

Find the integral to the curve \$[(2 - 1/{x^2})^2]\$ at the point (1, 0).

\$y = ∫[(2 - 1/{x^2})^2]dx = ∫[(4 - 4/{x^2} + 1/{x^4})]dx \$
\$ = 4x + 4/x - 1/{3x^{3}} + c\$
When \$x = 1\$ and \$y = 0\$, \$4 + 4 - 1/3 + c = 0\$, so \$c = -{23}/3\$
∴ \$y = 4x + 4/x - 1/{3x^{3}} - {23}/3\$

## Compound formula

Linear functions may be solved by the compound formula:

\$\$∫(ax + b)^n dx = 1/{a(n + 1)}⋅(ax + b)^{n+1} + c ,a ≠ 0\$\$

Associated Mathematicians:

• Isaac Newton
• Gottfried Leibnitz
• Jakob Bernoulli
• Johann Bernoulli (I)

Content © Andrew Bone. All rights reserved. Created : December 18, 2013 Last updated :December 13, 2015

## Site Index

### Latest Item on Science Library:

The most recent article is:

Air Resistance and Terminal Velocity

View this item in the topic:

Mechanics

and many more articles in the subject:

### Physics

Physics is the science of the very small and the very large. Learn about Isaac Newton, who gave us the laws of motion and optics, and Albert Einstein, who explained the relativity of all things, as well as catch up on all the latest news about Physics, on ScienceLibrary.info.

### Great Scientists

#### Wallace Carothers

1896 - 1937

Wallace Carothers was an American chemist and pioneer in pure research into large-molecular weight polymers.

### Quote of the day...

The truth about the climate crisis is an inconvenient one that means we are going to have to change the way we live our lives.