«K12/K13» 117. Hausaufgabe «PDF», «POD»

0.0.1 ↑ 117. Hausaufgabe

0.0.1.1 ↑ Analysis-Buch Seite 257, Aufgabe 19

Berechne:

f)

\int \frac{x^2}{\sqrt{1 - x^3}} \,\mathrm{d}x = -\frac{1}{3} \int \frac{1}{\sqrt{1 - x^3}} \cdot \left(1 - x^3\right)' \mathrm{d}x = -\frac{1}{3} \int \frac{1}{\sqrt{t}} \mathrm{d}t = -\frac{2}{3} \sqrt{1 - x^3} + C;$\int \; <!--nolimits-->\frac{x^2}{\sqrt{1-x^3}}\text{d}x=-\frac{1}{3}\int \; <!--nolimits-->\frac{1}{\sqrt{1-x^3}}\cdot {\left(1-x^3\right)}^{\prime }\text{d}x=-\frac{1}{3}\int \; <!--nolimits-->\frac{1}{\sqrt{t}}\text{d}t=-\frac{2}{3}\sqrt{1-x^3}+C;$

g)

\int \frac{\tan^3 x}{\cos^2 x} \,\mathrm{d}x = \int \tan^3 x \cdot \left(\tan x\right)' \mathrm{d}x = \int t^3 \,\mathrm{d}t = \frac{1}{4} \tan^4 x + C;$\int \; <!--nolimits-->\frac{{tan}^{3}x}{{cos}^{2}x}\text{d}x=\int \; <!--nolimits-->{tan}^{3}x\cdot {\left(tanx\right)}^{\prime }\text{d}x=\int \; <!--nolimits-->t^3\text{d}t=\frac{1}{4}{tan}^{4}x+C;$

h)

\int \frac{x}{1 + x^4} \,\mathrm{d}x$\int \; <!--nolimits-->\frac{x}{1+x^4}\text{d}x$