Ubestemt integral med delvis integrasjon
Oppgave
Bestem integralet
\[\int 4x \cdot \cos x \, dx \]
Fasit
\(4x\sin x + 4\cos x + C\)
Løsningsforslag
Vi bruker delvis integrasjon (DI-metoden):
| D | I | |
|---|---|---|
| \(+\) | \(\textcolor{seagreen}{4x}\) | \(\textcolor{seagreen}{\cos x}\) |
| \(-\) | \(\textcolor{tomato}{4}\) | \(\textcolor{tomato}{\sin x}\) |
| \(+\) | \(\textcolor{maroon}{0}\) | \(\textcolor{steelblue}{-\cos x}\) |
\[\int 4x \cdot \cos x \, \mathrm{d}x = \textcolor{seagreen}{4x}\textcolor{tomato}{\sin x} - \textcolor{tomato}{4} \textcolor{steelblue}{\left( - \cos x \right)} + \textcolor{maroon}{0} + C=4x \sin x + 4 \cos x +C \]
$$\underline{\underline{ 4(x \sin x + \cos x) + C }}$$