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Lösung 2.1:2a

Aus Online Mathematik Brückenkurs 2

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K
K (Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel))
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The integrand in our case consists of two terms in the form <math>x^n</math>, and so we can use the rule
The integrand in our case consists of two terms in the form <math>x^n</math>, and so we can use the rule
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{{Displayed math||<math>\int x^n\,dx = \frac{x^{n+1}}{n+1}+C</math>}}
+
{{Abgesetzte Formel||<math>\int x^n\,dx = \frac{x^{n+1}}{n+1}+C</math>}}
on the terms individually to obtain that
on the terms individually to obtain that
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{{Displayed math||<math>F(x) = \frac{x^{2+1}}{2+1} + 3\cdot \frac{x^{3+1}}{3+1}</math>}}
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{{Abgesetzte Formel||<math>F(x) = \frac{x^{2+1}}{2+1} + 3\cdot \frac{x^{3+1}}{3+1}</math>}}
is a primitive function of the integrand.
is a primitive function of the integrand.
Zeile 13: Zeile 13:
The integrand's value is thus
The integrand's value is thus
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{{Displayed math||<math>\begin{align}
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{{Abgesetzte Formel||<math>\begin{align}
\int\limits_{0}^{2} \bigl( x^2+3x^3\bigr)\,dx
\int\limits_{0}^{2} \bigl( x^2+3x^3\bigr)\,dx
&= \Bigl[\ \frac{x^3}{3} + 3\cdot\frac{x^4}{4}\Bigr]_0^2\\
&= \Bigl[\ \frac{x^3}{3} + 3\cdot\frac{x^4}{4}\Bigr]_0^2\\
Zeile 24: Zeile 24:
Note: One way to check that <math>F(x) = \tfrac{1}{3}x^3 + \tfrac{3}{4}x^4</math> is a primitive function of the integral is to differentiate <math>F(x)</math> and to see that we obtain
Note: One way to check that <math>F(x) = \tfrac{1}{3}x^3 + \tfrac{3}{4}x^4</math> is a primitive function of the integral is to differentiate <math>F(x)</math> and to see that we obtain
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{{Displayed math||<math>\begin{align}
+
{{Abgesetzte Formel||<math>\begin{align}
F'(x) &= \tfrac{1}{3}\bigl(x^3\bigr)' + \tfrac{3}{4}\bigl(x^4\bigr)'\\[5pt]
F'(x) &= \tfrac{1}{3}\bigl(x^3\bigr)' + \tfrac{3}{4}\bigl(x^4\bigr)'\\[5pt]
&= \tfrac{1}{3}\cdot 3x^2 + \tfrac{3}{4}\cdot 4x^3\\[5pt]
&= \tfrac{1}{3}\cdot 3x^2 + \tfrac{3}{4}\cdot 4x^3\\[5pt]

Version vom 12:58, 10. Mär. 2009

The foremost difficulty with calculating an integral is finding a primitive function of the integrand. Once we have done that, the integral is calculated as the difference between the primitive function's values in the upper and lower limits of integration.

The integrand in our case consists of two terms in the form xn, and so we can use the rule

xndx=xn+1n+1+C 

on the terms individually to obtain that

F(x)=x2+12+1+3x3+13+1

is a primitive function of the integrand.

The integrand's value is thus

20x2+3x3dx= 3x3+34x420=323+3424−303+3404=38+4316=344.


Note: One way to check that F(x)=31x3+43x4 is a primitive function of the integral is to differentiate F(x) and to see that we obtain

F(x)=31x3+43x4=313x2+434x3=x2+3x3

as the integrand.

Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_2.1:2a“