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L'intigrazzioni n'aiuta quannu circamu di [[murtipricazzioni|murtipricari]] unitati nta nu prubblema.<br>
Pa isempiu, into a problem. For example, if a problem with [[rate]] (<math>distance \over time</math>) needs an answer with just distance, one solution is to integrate with respect to time. This means multiplying in time (to cancel the time in <math>distance \over time</math>). This is done by adding small slices of the rate graph together. The slices are close to zero in width, but adding them forever makes them add up to a whole. This is called a [[Riemann Sum]].
Adding these slices together gives the [[equation]] that the first equation is the derivative of. Integrals are kind of like adding machines.
Another time integration is helpful is when finding the [[volume]] of a solid. It can add [[2D|two-dimensional]] (without width) slices of the solid together forever until there is a width. This means the object now has three dimensions: the original two and a width. This gives the volume of the [[three-dimensional]] object described.
==Methods of Integration==
===Simple Equations===
A simple equation such as <math>y = x^2</math> can be integrated with respect to x using the following technique. To integrate, you add 1 to the power x is raised to, and then divide x by the value of this new power. Therefore, integration of a normal equation follows the "mantra":
<math>\int_{\,}^{\,} x^n dx = \frac{x^{n+1}}{n+1} + C</math>
This can be seen to be the inverse of [[differentiation]]. However, there is a constant, C, added when you integrate. This is called the constant of integration. This is required because differentiating an integer results in [[zero]], therefore integrating zero (which can be put onto the end of any integrand) produces an integer, C. The value of this integer would be found by using given conditions.
Equations with more than one terms are simply integrated by integrating each individual term:
<math>\int_{\,}^{\,} x^2 + 3x - 2 dx = \int_{\,}^{\,} x^2 dx + \int_{\,}^{\,} 3x dx - \int_{\,}^{\,} 2 dx = \frac{x^3}{3} + \frac{3x^2}{2} - 2x + C</math>
===Integration involving e and ln===
There are certain rules for integrating using [[e]] and the natural logarithm. Most importantly, <math>e^x</math> is the integral of itself (with the addition of a constant of integration):
<math>\int_{\,}^{\,}e^{x} dx = e^{x} + C</math>
With e raised to a function of x (<math>f(x)</math>):
<math>\int_{\,}^{\,} e^{f(x)} dx = e^{f(x)}\div\int_{\,}^{\,}f(x) dx</math>
The natural logarithm, ln, is useful when integrating equations with <math>1/x</math>. These cannot be integrated using the formula above (add one to the power, divide by the power), because adding one to the power produces 0, and a division by 0 is not possible. Instead, the integral of <math>1/x</math> is <math>\ln x</math>:
<math>\int_{\,}^{\,}\frac{1}{x} dx = \ln x + C </math>
In a more general form:
<math>\int_{\,}^{\,}\frac{f'(x)}{f(x)} dx = \ln {|f(x)|} + C </math>
The two vertical bars indicated a [[absolute value]]; the sign (positive or negative) of <math>f(x)</math> is ignored. This is because there is no value for the natural logarithm of negative numbers.
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== Taliati macari ==
* [[Ntigrali di Riemann]]
[[Category:Mathematics]]
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