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Di Sep 20 17:05:24 CEST 2005

=head2 4. Hausaufgabe

=head3 Analysis-Buch Seite 15, Aufgabe 9

Zeige die Richtigkeit von

=over

=item a)

M<\int \left(x^2 - x\right) \mathrm{d}x = \frac{1}{3}x^3 - \frac{1}{2}x^2 + C;>

M<\left(\frac{1}{3}x^3 - \frac{1}{2}x^2 + C\right)' = x^2 - x;>

=item b)

M<\int \sqrt{x} \,\mathrm{d}x = \frac{2}{3}x\sqrt{x} + C;>

M<\left(\frac{2}{3}x\sqrt{x} + C\right)' =
\left(\frac{2}{3}x^{\frac{3}{2}} + C\right)' = x^{\frac{1}{2}} = \sqrt{x};>

=item c)

M<\int \frac{1}{x^2} \,\mathrm{d}x = -\frac{1}{x} + C;>

M<\left(-\frac{1}{x} + C\right)' = \frac{1}{2}x^2;>

=item d)

M<\int \sin^2 x \,\mathrm{d}x = \frac{1}{2}\left(x - \sin x \cos x\right) + C;>

M<\left[\frac{1}{2}\left(x - \sin x \cos x\right) + C\right]' =
\frac{1}{2}\left(1 - \cos x \cos x + \sin x \sin x\right) = \frac{1}{2}\left(1
- 1 + \sin^2 x + \sin^2 x\right) = \sin^2 x;>

=item e)

M<\int \cos^2 x \,\mathrm{d}x = \frac{1}{2}\left(x + \sin x \cos x\right) + C;>

M<\left[\frac{1}{2}\left(x + \sin x \cos x\right) + C\right]' =
\frac{1}{2}\left(1 + \cos x \cos x - \sin x \sin x\right) = \frac{1}{2}\left(1
- 1 + \cos^2 x + \cos^2 x\right) = \cos^2 x;>

=item f)

M<\int \frac{x}{\sqrt{a^2 - x^2}} \,\mathrm{d}x = -\sqrt{a^2 - x^2} + C;>

M<\left(-\sqrt{a^2 - x^2} + C\right)' = -\dfrac{1}{2\sqrt{a^2 - x^2}}\left(0 -
2x\right) = \dfrac{x}{\sqrt{a^2 - x^2}};>

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