$$\int a\cdot x^n \ dx = \frac{a}{n+1}\cdot x^{n+1}+C$$

Oppgave 1

  • \begin{align*}\int e^x \ dx&=e^x +C\end{align*}

  • \begin{align*}\int x \ dx&=\frac{1}{2}x^2+C\end{align*}

  • \begin{align*}\int 2 \ dx&=2x+C\end{align*}

  • \begin{align*}\int x^2 \ dx&=\frac{1}{3}x^3+C\end{align*}

Oppgave 2

  • \begin{align*}\int \frac{1}{x} \ dx&=\ln(x)+C\end{align*}

  • \begin{align*}\int x^5 \ dx&=\frac{1}{6}x^6+C\end{align*}

  • \begin{align*}\int 3x^2 \ dx&=\frac{3}{3}x^3+C\\&=x^3+C\end{align*}

  • \begin{align*}\int x^2+x+1 \ dx&=\frac{1}{3}x^3+\frac{1}{2}x^2+x+C\end{align*}

Oppgave 3

  • \begin{align*}\int e^{2x} \ dx&=\frac{1}{2}e^{2x}+C\end{align*}

  • \begin{align*}\int \frac{1}{x^2} \ dx&=\int x^{-2} \ dx\\&= \frac{1}{-2+1}x^{-2+1}+C\\&=-x^{-1}+C\\&=-\frac{1}{x}+C\end{align*}

  • \begin{align*}\int \pi+x \ dx&=\pi x +\frac{1}{2}x^2+C\end{align*}

  • \begin{align*}\int 2e+e^x \ dx&=2 e x+e^x+C\end{align*}

Oppgave 4

  • \begin{align*}\int 3x^4 \ dx&=\frac{3}{4+1}x^{4+1}+C\\&=\frac{3}{5}x^5+C\end{align*}

  • \begin{align*}\int \frac{1}{3}x^3+ \frac{1}{2}x^2+x+1 \ dx&=\frac{1}{12}x^4+\frac{1}{6}x^3+\frac{1}{2}x^2+x+C\end{align*}

  • \begin{align*}\int -x^{-3} \ dx&=-(\frac{1}{-3+1})x^{-3+1}+C\\&=\frac{1}{2}x^{-2}\\&=\frac{1}{2x^2}+C\end{align*}

  • \begin{align*}\int \sqrt{x} \ dx &=\int x^{\frac{1}{2}} \ dx\\&=\frac{1}{\frac{1}{2}+1}x^{\frac{1}{2}+1}+C\\& =\frac{2}{3}x\sqrt{x}+C\end{align*}

Oppgave 5

  • \begin{align*}\int (2x+2) \ dx&=x^2+2x+C\end{align*}

  • \begin{align*}\int \sqrt[3]{x} \ dx&=\int x^{\frac{1}{3}}\ dx\\&=\frac{1}{\frac{1}{3}+1}\cdot x^{\frac{1}{3}+1}+C\\&=\frac{1}{\frac{4}{3}}\cdot x^{\frac{4}{3}}+C\\&=\frac{3}{4}\cdot \sqrt[3]{x^4}+C\\&=\frac{3}{4}\cdot \sqrt[3]{x^3}\cdot\sqrt[3]{x}+C\\&=\frac{3}{4}\cdot x\cdot\sqrt[3]{x}+C\\\end{align*}

  • \begin{align*}\int 2e^x \ dx=2e^x+C\end{align*}

  • \begin{align*}\int e^{3x}+e^x+e^2 \ dx=\frac{1}{3}\cdot e^{3x}+e^x+e^2\cdot x+C\end{align*}

  • \begin{align*}\int 2^x \ dx=\frac{2^x}{\ln 2}+C\end{align*}

Oppgave 6

  • \begin{align*}\int \ln{3}\cdot 3^x \ dx&= \ln{3}\int 3^x \ dx\\&=\ln{3}\cdot \frac{1}{\ln 3} \cdot 3^x+C\\&=3^x+C\end{align*}

  • \begin{align*}\int 3\sin{x}+3 \ dx=-3 \cos x +3x+C\end{align*}

  • \begin{align*}\int \cos{x}-5x \ dx=\sin x-\frac{5}{2}x^2+C\end{align*}

  • \begin{align*}\int e^x -\cos{x}\ dx=e^x-\sin x+C\end{align*}