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 © Renewable.Media. All rights reserved. Created : December 18, 2013 Last updated :December 13, 2015
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Mathematics
Mathematics is the most important tool of science. The quest to understand the world and the universe using mathematics is as old as civilisation, and has led to the science and technology of today. Learn about the techniques and history of mathematics on ScienceLibrary.info.
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Johann(I) Bernoulli
1667 - 1748
Johann Bernoulli was a Swiss mathematician, one of two brothers who founded the Bernoulli family dynasty which dominated mathematics for more than a century.
_portrait.jpg "Johann Bernoulli, 1667 - 1748, Swiss mathematician")
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