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Lösung 2.2:3e

Aus Online Mathematik Brückenkurs 2

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K (Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel))
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If we differentiate the denominator in the integrand
If we differentiate the denominator in the integrand
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{{Displayed math||<math>(x^2+1)' = 2x</math>}}
+
{{Abgesetzte Formel||<math>(x^2+1)' = 2x</math>}}
we obtain almost the same expression as in the numerator; there is a constant 2 which is different. We therefore rewrite the numerator as
we obtain almost the same expression as in the numerator; there is a constant 2 which is different. We therefore rewrite the numerator as
-
{{Displayed math||<math>3x = \frac{3}{2}\cdot 2x = \frac{3}{2}\cdot (x^2+1)',</math>}}
+
{{Abgesetzte Formel||<math>3x = \frac{3}{2}\cdot 2x = \frac{3}{2}\cdot (x^2+1)',</math>}}
so the integral can be written as
so the integral can be written as
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{{Displayed math||<math>\int\frac{\tfrac{3}{2}}{x^2+1}\cdot (x^{2}+1)'\,dx\,,</math>}}
+
{{Abgesetzte Formel||<math>\int\frac{\tfrac{3}{2}}{x^2+1}\cdot (x^{2}+1)'\,dx\,,</math>}}
and we see that the substitution <math>u=x^2+1</math> can be used to simplify the integral,
and we see that the substitution <math>u=x^2+1</math> can be used to simplify the integral,
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{{Displayed math||<math>\begin{align}
+
{{Abgesetzte Formel||<math>\begin{align}
\int \frac{3x}{x^2+1}\,dx
\int \frac{3x}{x^2+1}\,dx
&= \left\{ \begin{align}
&= \left\{ \begin{align}

Version vom 13:02, 10. Mär. 2009

If we differentiate the denominator in the integrand

(x2+1)=2x

we obtain almost the same expression as in the numerator; there is a constant 2 which is different. We therefore rewrite the numerator as

3x=232x=23(x2+1)

so the integral can be written as

23x2+1(x2+1)dx 

and we see that the substitution u=x2+1 can be used to simplify the integral,

3xx2+1dx=udu=x2+1=(x2+1)dx=2xdx=23udu=23lnu+C=23lnx2+1+C=23ln(x2+1)+C.

In the last step, we take away the absolute sign around the argument in ln, because x2+1 is always greater than or equal to 1.

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