Integralen av $$$x^{66} \tan{\left(1 \right)} + x^{2}$$$

Bestäm $$$\int \left(x^{66} \tan{\left(1 \right)} + x^{2}\right)\, dx$$$.

Integrera termvis:

$${\color{red}{\int{\left(x^{66} \tan{\left(1 \right)} + x^{2}\right)d x}}} = {\color{red}{\left(\int{x^{2} d x} + \int{x^{66} \tan{\left(1 \right)} d x}\right)}}$$

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

$$\int{x^{66} \tan{\left(1 \right)} d x} + {\color{red}{\int{x^{2} d x}}}=\int{x^{66} \tan{\left(1 \right)} d x} + {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\int{x^{66} \tan{\left(1 \right)} d x} + {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\tan{\left(1 \right)}$$$ och $$$f{\left(x \right)} = x^{66}$$$:

$$\frac{x^{3}}{3} + {\color{red}{\int{x^{66} \tan{\left(1 \right)} d x}}} = \frac{x^{3}}{3} + {\color{red}{\tan{\left(1 \right)} \int{x^{66} d x}}}$$

Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=66$$$:

$$\frac{x^{3}}{3} + \tan{\left(1 \right)} {\color{red}{\int{x^{66} d x}}}=\frac{x^{3}}{3} + \tan{\left(1 \right)} {\color{red}{\frac{x^{1 + 66}}{1 + 66}}}=\frac{x^{3}}{3} + \tan{\left(1 \right)} {\color{red}{\left(\frac{x^{67}}{67}\right)}}$$

Alltså,

$$\int{\left(x^{66} \tan{\left(1 \right)} + x^{2}\right)d x} = \frac{x^{67} \tan{\left(1 \right)}}{67} + \frac{x^{3}}{3}$$

Lägg till integrationskonstanten:

$$\int{\left(x^{66} \tan{\left(1 \right)} + x^{2}\right)d x} = \frac{x^{67} \tan{\left(1 \right)}}{67} + \frac{x^{3}}{3}+C$$

$$$\int \left(x^{66} \tan{\left(1 \right)} + x^{2}\right)\, dx = \left(\frac{x^{67} \tan{\left(1 \right)}}{67} + \frac{x^{3}}{3}\right) + C$$$A