Exercise 3 | Wiskunde op Tilburg University

Determine the value of the integral

$\int_0^\infty (\frac{1}{2})^x dx$ .

Antwoord 1 correct

Correct

Antwoord 2 optie

$0$

Antwoord 2 correct

Fout

Antwoord 3 optie

$-1$

Antwoord 3 correct

Fout

Antwoord 4 optie

$\infty$

Antwoord 4 correct

Fout

Antwoord 1 optie

$1/\ln 2$

Antwoord 1 feedback

Correct:

$\int_0^t (\frac{1}{2})^x dx=[(\frac{1}{2})^x/\ln\frac{1}{2}]_{x=0}^{x=t}=(1-(\frac{1}{2})^t)/\ln 2$ . It holds that

$(1-(\frac{1}{2})^t)/\ln 2\rightarrow 1/\ln 2$ if

$t\rightarrow\infty$ .

Antwoord 2 feedback

Wrong: Note that

$(\frac{1}{2})^t\rightarrow 0$ if

$t\rightarrow\infty$ .

Try again.

Antwoord 3 feedback

Wrong: Note that

$(\frac{1}{2})^x$ is not an antiderivative of

$(\frac{1}{2})^x$ .

See Antidifferentiation.

Antwoord 4 feedback

Wrong: What happens with the term

$(\frac{1}{2})^x$ if

$x$ becomes large?

See Improper integral (film).