Lösung 2.2:3a

Aus Online Mathematik Brückenkurs 2

(Unterschied zwischen Versionen)
Wechseln zu: Navigation, Suche
K
K (Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel))
Zeile 1: Zeile 1:
The secret behind a successful substitution is to be able to recognize the integral as an expression of the type
The secret behind a successful substitution is to be able to recognize the integral as an expression of the type
-
{{Displayed math||<math>\int \left( \begin{matrix}
+
{{Abgesetzte Formel||<math>\int \left( \begin{matrix}
\text{an expression}\\
\text{an expression}\\
\text{in u}
\text{in u}
Zeile 8: Zeile 8:
where <math>u=u(x)</math> is the actual substitution. In the integral
where <math>u=u(x)</math> is the actual substitution. In the integral
-
{{Displayed math||<math>\int 2x\sin x^2\,dx</math>}}
+
{{Abgesetzte Formel||<math>\int 2x\sin x^2\,dx</math>}}
we see that the expression <math>x^2</math> is the argument for the sine function, as the same time as its derivative <math>\bigl(x^2\bigr)'=2x</math> stands as a factor in front of sine. Therefore, if we set <math>u=x^2</math>, the integral, the integral will be of the form
we see that the expression <math>x^2</math> is the argument for the sine function, as the same time as its derivative <math>\bigl(x^2\bigr)'=2x</math> stands as a factor in front of sine. Therefore, if we set <math>u=x^2</math>, the integral, the integral will be of the form
-
{{Displayed math||<math>\int u'\sin u\,dx\,\textrm{.}</math>}}
+
{{Abgesetzte Formel||<math>\int u'\sin u\,dx\,\textrm{.}</math>}}
Thus, we can use <math>u=x^2</math> for the substitution,
Thus, we can use <math>u=x^2</math> for the substitution,
-
{{Displayed math||<math>\begin{align}
+
{{Abgesetzte Formel||<math>\begin{align}
\int 2x\sin x^2\,dx
\int 2x\sin x^2\,dx
&=\left\{\begin{align}
&=\left\{\begin{align}

Version vom 13:01, 10. Mär. 2009

The secret behind a successful substitution is to be able to recognize the integral as an expression of the type

\displaystyle \int \left( \begin{matrix}

\text{an expression}\\ \text{in u} \end{matrix}\right)\cdot u'\,dx\,,

where \displaystyle u=u(x) is the actual substitution. In the integral

\displaystyle \int 2x\sin x^2\,dx

we see that the expression \displaystyle x^2 is the argument for the sine function, as the same time as its derivative \displaystyle \bigl(x^2\bigr)'=2x stands as a factor in front of sine. Therefore, if we set \displaystyle u=x^2, the integral, the integral will be of the form

\displaystyle \int u'\sin u\,dx\,\textrm{.}

Thus, we can use \displaystyle u=x^2 for the substitution,

\displaystyle \begin{align}

\int 2x\sin x^2\,dx &=\left\{\begin{align} u &= x^2\\[5pt] du &= 2x\,dx \end{align}\right\} = \int{\sin u\,du}\\[5pt] &= -\cos u+C = -\cos x^2 + C\,\textrm{.} \end{align}

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_2.2:3a“