Lösung 2.1:3d
Aus Online Mathematik Brückenkurs 2
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By dividing the two terms in the numerator by <math>x</math>, we can simplify each term to a form which makes it possible simply to write down the primitive functions of the integrand, | By dividing the two terms in the numerator by <math>x</math>, we can simplify each term to a form which makes it possible simply to write down the primitive functions of the integrand, | ||
| - | {{ | + | {{Abgesetzte Formel||<math>\begin{align} |
\int \frac{x^{2}+1}{x}\,dx | \int \frac{x^{2}+1}{x}\,dx | ||
&= \int \Bigl(\frac{x^2}{x} + \frac{1}{x}\Bigr)\,dx\\[5pt] | &= \int \Bigl(\frac{x^2}{x} + \frac{1}{x}\Bigr)\,dx\\[5pt] | ||
Version vom 12:59, 10. Mär. 2009
By dividing the two terms in the numerator by \displaystyle x, we can simplify each term to a form which makes it possible simply to write down the primitive functions of the integrand,
| \displaystyle \begin{align}
\int \frac{x^{2}+1}{x}\,dx &= \int \Bigl(\frac{x^2}{x} + \frac{1}{x}\Bigr)\,dx\\[5pt] &= \int \bigl(x+x^{-1}\bigr)\,dx\\[5pt] &= \frac{x^2}{2} + \ln |x| + C\,, \end{align} |
where \displaystyle C is an arbitrary constant.
Note: Observe that \displaystyle 1/x has a singularity at \displaystyle x=0, so the answers above are only primitive functions over intervals that do not contain \displaystyle x=0\,.
