Integralen av $$$\frac{5 x^{6} + 5}{x^{2} + 1}$$$

Bestäm $$$\int \frac{5 x^{6} + 5}{x^{2} + 1}\, dx$$$.

Förenkla integranden:

$${\color{red}{\int{\frac{5 x^{6} + 5}{x^{2} + 1} d x}}} = {\color{red}{\int{\left(5 x^{4} - 5 x^{2} + 5\right)d x}}}$$

Integrera termvis:

$${\color{red}{\int{\left(5 x^{4} - 5 x^{2} + 5\right)d x}}} = {\color{red}{\left(\int{5 d x} - \int{5 x^{2} d x} + \int{5 x^{4} d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=5$$$:

$$- \int{5 x^{2} d x} + \int{5 x^{4} d x} + {\color{red}{\int{5 d x}}} = - \int{5 x^{2} d x} + \int{5 x^{4} d x} + {\color{red}{\left(5 x\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=5$$$ och $$$f{\left(x \right)} = x^{2}$$$:

$$5 x + \int{5 x^{4} d x} - {\color{red}{\int{5 x^{2} d x}}} = 5 x + \int{5 x^{4} d x} - {\color{red}{\left(5 \int{x^{2} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=2$$$:

$$5 x + \int{5 x^{4} d x} - 5 {\color{red}{\int{x^{2} d x}}}=5 x + \int{5 x^{4} d x} - 5 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=5 x + \int{5 x^{4} d x} - 5 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=5$$$ och $$$f{\left(x \right)} = x^{4}$$$:

$$- \frac{5 x^{3}}{3} + 5 x + {\color{red}{\int{5 x^{4} d x}}} = - \frac{5 x^{3}}{3} + 5 x + {\color{red}{\left(5 \int{x^{4} d x}\right)}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=4$$$:

$$- \frac{5 x^{3}}{3} + 5 x + 5 {\color{red}{\int{x^{4} d x}}}=- \frac{5 x^{3}}{3} + 5 x + 5 {\color{red}{\frac{x^{1 + 4}}{1 + 4}}}=- \frac{5 x^{3}}{3} + 5 x + 5 {\color{red}{\left(\frac{x^{5}}{5}\right)}}$$

Alltså,

$$\int{\frac{5 x^{6} + 5}{x^{2} + 1} d x} = x^{5} - \frac{5 x^{3}}{3} + 5 x$$

Förenkla:

$$\int{\frac{5 x^{6} + 5}{x^{2} + 1} d x} = x \left(x^{4} - \frac{5 x^{2}}{3} + 5\right)$$

Lägg till integrationskonstanten:

$$\int{\frac{5 x^{6} + 5}{x^{2} + 1} d x} = x \left(x^{4} - \frac{5 x^{2}}{3} + 5\right)+C$$

$$$\int \frac{5 x^{6} + 5}{x^{2} + 1}\, dx = x \left(x^{4} - \frac{5 x^{2}}{3} + 5\right) + C$$$A