Quiz about Analysis I - Übungsstunde 13

Question 1

Was ist eine Parametrisierung $$\left[-r,r\right]\rightarrow\ \gamma$$ von der Halbkreis $$\gamma$$ mit Radius $$r$$

Accepted Answer

$$\gamma\left(x\right)\ =\ \sqrt{r^2-x^2}$$

Answer

$$\gamma\left(x\right)\ =\ \sqrt{x^2-r^2}$$

Answer

$$\gamma\left(x\right)\ =\ x^2-r^2$$

Answer

$$\gamma\left(x\right)\ =\ r^2-x^2$$

Question 2

Bestimme den Wert von $$I=\int\ \frac{1}{x^2-1}dx$$ mittels PBZ

Accepted Answer

$$I=\frac{1}{2}\ln\left|x-1\right|-\frac{1}{2}\ln\left|x+1\right|$$

Answer

$$I=\frac{1}{2}\ln\left|x+1\right|-\frac{1}{2}\ln\left|x-1\right|$$

Answer

$$I=2\ln\left|x-1\right|-2\ln\left|x+1\right|$$

Answer

$$I=2\ln\left|x+1\right|-2\ln\left|x-1\right|$$

Question 3

$$\int_a^b\ \frac{1}{1-x^2}dx=-\int_c^d\ \frac{1}{\sin\left(t\right)}dt$$ mittels die Substitution $$x=\cos\left(t\right)$$ . Was sind $$c$$ und $$d$$ ?

Accepted Answer

$$c\ =\arccos\left(a\right),$$

$$d\ =\arccos\left(b\right)$$

Answer

$$c\ =\cos\left(a\right),\ d=\cos\left(b\right)$$

Answer

$$c\ =-\sin\left(a\right),\ d=-\sin\left(b\right)$$

Answer

$$c\ =-\arcsin\left(a\right),$$

$$d\ =-\arcsin\left(b\right)$$

Question 4

$$\int\ln\left(x\right)dx\ =?$$ Hinweis: $$f\left(x\right)=\ln\left(x\right),\ g$$

Accepted Answer

$$x\ln\left(x\right)-x+C$$

Answer

$$x\ln\left(x\right)$$

Answer

$$x\ln\left(x\right)-x$$

Answer

$$\ln\left(x\right)-x+C$$

Question 5

$$\int\ \frac{1}{1+x^2}dx\ =\ ?$$

Accepted Answer

$$\arctan\left(x\right)+C$$

Answer

$$\arcsin\left(x\right)+C$$

Answer

$$\operatorname{arccosh}\left(x\right)+C$$

Answer

$$\arccos\left(x\right)+C$$

Question 6

Sei $$I\ =\ \int\ f\left(\ln\left(x\right)\right)dx$$ . Dann ist $$I\ =\ ?$$

Accepted Answer

$$\int\ f\left(t\right)e^tdt$$

Answer

$$\int f\left(\ln\left(t\right)\right)e^tdt$$

Answer

$$\int f\left(t\right)\frac{1}{t}dt$$

Answer

$$\int f\left(t\right)\ln\left(t\right)dt$$

Question 7

$$\int_0^1x^{\frac{3}{5}}dx\ =?$$

Accepted Answer

$$\frac{5}{8}$$

Answer

$$\frac{5}{3}$$

Answer

$$-\frac{5}{2}$$

Answer

$$\frac{3}{5}$$

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