Eksempel 1

Oppgave 2

Regn ut integralet $\displaystyle \int \left(-2x+\frac{7}{x^3} + \frac{1}{4x^{\frac{1}{3}}}\right) dx$.

Løsning

Vi integrerer hvert ledd hver for seg der vi setter konstanter foran integrasjonsleddet:

$$\begin{align*} &\int \left(-2x+\frac{7}{x^3} + \frac{1}{4x^{\frac{1}{3}}}\right) = \int -2x \ dx + \int \frac{7}{x^3} dx + \int \frac{1}{4x^{\frac{1}{3}}} dx \\ &= -2\int x \ dx + 7\int x^{-3} \ dx + \frac{1}{4}\int x^{-\frac{1}{3}} \ dx \\ &= -2\cdot\frac{1}{2}x^2 + 7\cdot\frac{1}{-2}x^{-2} + \frac{1}{4}\cdot\frac{1}{-\frac{1}{3}+1}x^{-\frac{1}{3}+1} + C \\ &= -x^2 - \frac{7}{2x^2} + \frac{1}{4}\cdot\frac{1}{\frac{2}{3}}x^{\frac{2}{3}} + C \\ &= \underline{\underline{-x^2 -\frac{7}{2x^2} + \frac{3}{8}x^{\frac{2}{3}} + C}} \end{align*}$$