Title: Määrätty integraali

Käyrän \(y=e^{[value]x}\) ja x-akselin väliin jäävän alueen pinta-ala on [value] välillä \([0,r]\). Määritä päätepiste \(r\) yhden desimaalin tarkkuudella.

\(r=\)
Feedback: Integroidaan välillä \([0,r]\) \[\int_{0}^{r}e^{[value]x}dx={1\over[value]}\int_{0}^{r}[value]\cdot e^{[value]x}dx={1\over[value]}\Bigl/_{\!\!\!0}^{r}e^{[value]x}={1\over[value]}(e^{[value]r}-e^{0})\] Integraalin on saatava arvo \({1\over[value]}(e^{[value]r}-e^{0})=[value]\quad||\cdot{1\over[value]}\) \(e^{[value]r}=[value]+1\quad||ln()\) \([value]r=ln([value]+1)\quad||:[value]\) \(r=\dfrac{ln([value]+1)}[value]\approx[value]\)

Määritä vakio \(t\) \[\int_{\Large e^[value]}^{\Large e^{t}}\dfrac[value]{x}dx=[value]\]

\(t=\)
Feedback: Integroidaan \[\int_{\Large e^[value]}^{\Large e^{t}}\dfrac[value]{x}dx=[value]\int_{\Large e^[value]}^{\Large e^{t}}\dfrac{1}{x}dx=[value]\Bigl/_{\!\!\!\Large e^[value]}^{\Large e^{t}}ln(x)=[value](ln(e^{t})-ln(e^[value]))=[value](t-([value]))\] \([value](t-([value]))=[value]\), josta \(t=[value]\)

Määritä integraalin tarkka arvo

 \[\int_[value]^[value]e^{x}dx=\]
Feedback: \[\int_[value]^[value]e^{x}dx=\Bigl/_{\!\!\![value]}^[value]e^{x}=(e^[value]-e^[value])=[value]-[value]=[value]\]

Määritä integraalin arvo kahden merkitsevän numeron tarkkuudella.

 \[\int_[value]^[value]e^{[value]x}dx=\]
Feedback: \[\int_[value]^[value]e^{[value]x}dx={1\over[value]}\int_[value]^[value][value]\cdot e^{[value]x}dx={1\over[value]}\Bigl/_{\!\!\![value]}^[value]e^{[value]x}={1\over[value]}(e^[value]-e^[value])=[value]\approx[value]\]

Integroi \[\int_{\Large e^[value]}^{\Large e^[value]}\dfrac[value]{x}dx=\]
Feedback: \[\int_{\Large e^[value]}^{\Large e^[value]}\dfrac[value]{x}dx=[value]\int_{\Large e^[value]}^{\Large e^[value]}\dfrac{1}{x}dx=[value]\Bigl/_{\!\!\!\Large e^[value]}^{\Large e^[value]}ln(x)=[value](ln(e^[value])-ln(e^[value]))=[value]([value]-([value]))=[value]\]

Integroi \[\int_[value]^[value]([value]x^[value]+[value]x^[value])dx=\]
Feedback: \[\int_[value]^[value]([value]x^[value]+[value]x^[value])dx=\Bigl/_{\!\!\![value]}^[value]([value]x^[value]+[value]x^[value])=([value]\cdot[value]^[value]+[value]\cdot[value]^[value])-([value]\cdot([value])^[value]+[value]\cdot([value])^[value])=[value]\]