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Lösung 2.2:1a

Aus Online Mathematik Brückenkurs 2

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K (Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel))
Zeile 11: Zeile 11:
The relation between <math>dx</math> and <math>du</math> reads
The relation between <math>dx</math> and <math>du</math> reads
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{{Displayed math||<math>du = u'(x)\,dx = (3x-1)'\,dx = 3\,dx\,,</math>}}
+
{{Abgesetzte Formel||<math>du = u'(x)\,dx = (3x-1)'\,dx = 3\,dx\,,</math>}}
which means that <math>dx</math> is replaced by <math>\tfrac{1}{3}\,du</math>.
which means that <math>dx</math> is replaced by <math>\tfrac{1}{3}\,du</math>.
Zeile 20: Zeile 20:
One usually writes the whole substitution of variables as
One usually writes the whole substitution of variables as
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{{Displayed math||<math>\int\limits_1^2 \frac{dx}{(3x-1)^4} = \left\{ \begin{align}
+
{{Abgesetzte Formel||<math>\int\limits_1^2 \frac{dx}{(3x-1)^4} = \left\{ \begin{align}
u &= 3x-1\\[5pt]
u &= 3x-1\\[5pt]
du &= 3\,dx
du &= 3\,dx
Zeile 27: Zeile 27:
Sometimes, we are more brief and hide the details,
Sometimes, we are more brief and hide the details,
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{{Displayed math||<math>\int\limits_1^2 \frac{dx}{(3x-1)^4} = \bigl\{ u=3x-1 \bigr\} = \int\limits_2^5 \frac{\tfrac{1}{3}\,du}{u^4}\,\textrm{.}</math>}}
+
{{Abgesetzte Formel||<math>\int\limits_1^2 \frac{dx}{(3x-1)^4} = \bigl\{ u=3x-1 \bigr\} = \int\limits_2^5 \frac{\tfrac{1}{3}\,du}{u^4}\,\textrm{.}</math>}}
After the substitution of variables, we have a standard integral which is easy to compute.
After the substitution of variables, we have a standard integral which is easy to compute.
Zeile 33: Zeile 33:
In summary, the whole calculation is,
In summary, the whole calculation is,
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{{Displayed math||<math>\begin{align}
+
{{Abgesetzte Formel||<math>\begin{align}
\int\limits_1^2 \frac{dx}{(3x-1)^4}
\int\limits_1^2 \frac{dx}{(3x-1)^4}
&= \left\{\begin{align}
&= \left\{\begin{align}

Version vom 13:00, 10. Mär. 2009

A substitution of variables is often carried out so as to transform a complicated integral to one that is less complicated which one can either directly calculate or continue to work with.

When we carry out a substitution of variables u=u(x), there are three things which are affected in the integral:

  1. the integral must be rewritten in terms of the new variable u;
  2. the element of integration, dx, is replaced by du, according to the formula du=u(x)dx;
  3. the limits of integration are for x and must be changed to limits of integration for the variable u.

In this case, we will perform the change of variables u=3x−1, mainly because the integrand 1(3x−1)4 will then be replaced by 1u4.

The relation between dx and du reads

du=u(x)dx=(3x−1)dx=3dx

which means that dx is replaced by 31du.

Furthermore, when x=1 in the lower limit of integration, the corresponding u-value becomes u=31−1=2, and when x=2, we obtain the u-value u=32−1=5.

One usually writes the whole substitution of variables as

21dx(3x−1)4=udu=3x−1=3dx=52u431du. 

Sometimes, we are more brief and hide the details,

21dx(3x−1)4=u=3x−1=52u431du. 

After the substitution of variables, we have a standard integral which is easy to compute.

In summary, the whole calculation is,

21dx(3x−1)4=udu=3x−1=3dx=52u431du=3152u−4du=31 u−4+1−4+1 52=−91 1u3 52=−91153−123=−91235323−53=117322353=3213322353=132353=131000.
Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_2.2:1a“