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.
Try again.
Antwoord 3 feedback
Wrong: Note that $(\frac{1}{2})^x$ is not an antiderivative of $(\frac{1}{2})^x$.
See Antidifferentiation.
See Antidifferentiation.
Antwoord 4 feedback
Wrong: What happens with the term $(\frac{1}{2})^x$ if $x$ becomes large?
See Improper integral (film).
See Improper integral (film).