Title: Integroiminen

Integroi \(\Large x^[value]\cdot e^{x^[value]}\)
Feedback: \(\Large x^[value]\cdot e^{x^[value]}\) Muokataan kertoimeksi sisäfunktion derivaatta \[\int x^[value]\cdot e^{x^[value]}dx={1\over[value]}\int[value]\cdot x^[value]\cdot e^{x^[value]}dx={1\over[value]}\cdot e^{x^[value]}+C=\dfrac{e^{x^[value]}}[value]+C\]

Integroi \(e^{[value]x}\)
Feedback: \(e^{[value]x}\) Muokataan kertoimeksi sisäfunktion derivaatta \[\int e^{[value]x}dx={1\over[value]}\int[value]\cdot e^{[value]x}dx={1\over[value]}\cdot e^{[value]x}+C=\dfrac{e^{[value]x}}[value]+C\]

Integroi \([value]x+[value]\)
Feedback: \(\int([value]x+[value])dx=[value]x^2+[value]x+C\)

Määritä se funktion \(f(x)=\dfrac[value]{x}\) integraalifunktio, jolle \(F(e^[value])=[value]\), kun \(x>0\)

\(F(x)=\)
Feedback: \[\int f(x)dx=\int\dfrac[value]{x}dx=[value]\int\dfrac{1}{x}dx=[value]\cdot ln(x)+C\] Integraalifunktiot \(F(x)=[value]\cdot ln(x)+C\) Sijoitus \(F(e^[value])=[value]\cdot ln(e^[value])+C=[value]\), josta \(C=[value]\) Haettu integraalifunktio on \(F(x)=[value]\cdot ln(x)[value][value]\)

Integroi \(\dfrac{[value]x}{x^2+[value]}\), kun \(x>0\)
Feedback: \(\dfrac{[value]x}{x^2+[value]}\) Muokataan osoittajaan nimittäjän derivaatta \(\dfrac{[value]x}{x^2+[value]}=[value]\cdot\dfrac{2x}{x^2+[value]}\) Integroidaan \[\int\dfrac{[value]x}{x^2+[value]}dx=\int[value]\cdot\dfrac{2x}{x^2+[value]}dx=[value]\int\dfrac{2x}{x^2+[value]}dx=[value]\cdot ln(x^2+[value])+C\]

Integroi \(\dfrac[value]{x}\), kun \(x>0\)
Feedback: \[\int\dfrac[value]{x}dx=[value]\int\dfrac{1}{x}dx=[value]\cdot ln(x)+C\]

Integroi \([value]x^[value]+[value]x^[value]\)
Feedback: \(\int([value]x^[value]+[value]x^[value])dx=\dfrac[value][value]x^{[value]+1}+\dfrac[value][value]x^{[value]+1}+C=[value]x^[value]+[value]x^[value]+C\)

Integroi \([value]x^[value]\)
Feedback: \(\int[value]x^[value]dx=\dfrac[value][value]\cdot x^{[value]+1}+C=[value]\)