Processing Math: Done
To print higher-resolution math symbols, click the
Hi-Res Fonts for Printing button on the jsMath control panel.
jsMath

Lösung 2.3:2c

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:
If we use the definition of <math>\tan x</math> and write the integral as
If we use the definition of <math>\tan x</math> and write the integral as
-
{{Displayed math||<math>\int\tan x\,dx = \int\frac{\sin x}{\cos x}\,dx</math>}}
+
{{Abgesetzte Formel||<math>\int\tan x\,dx = \int\frac{\sin x}{\cos x}\,dx</math>}}
we see that the numerator <math>\sin x</math> is the derivative of the denominator (apart from the minus sign). Hence, the substitution <math>u=\cos x</math> will work,
we see that the numerator <math>\sin x</math> is the derivative of the denominator (apart from the minus sign). Hence, the substitution <math>u=\cos x</math> will work,
-
{{Displayed math||<math>\begin{align}
+
{{Abgesetzte Formel||<math>\begin{align}
\int\frac{\sin x}{\cos x}\,dx
\int\frac{\sin x}{\cos x}\,dx
&= \left\{\begin{align}
&= \left\{\begin{align}

Version vom 13:04, 10. Mär. 2009

If we use the definition of tanx and write the integral as

tanxdx=sinxcosxdx 

we see that the numerator sinx is the derivative of the denominator (apart from the minus sign). Hence, the substitution u=cosx will work,

sinxcosxdx=udu=cosx=(cosx)dx=−sinxdx=−udu=−lnu+C=−lncosx+C.


Note: −lncosx+C is only a primitive function in intervals in which cosx=0.

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