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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

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