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Integralen av $$$- \frac{8767 x^{2}}{10000} + \frac{8767 \sin{\left(x \right)}}{10000}$$$
Relaterad kalkylator: Kalkylator för bestämda och oegentliga integraler
Din inmatning
Bestäm $$$\int \left(- \frac{8767 x^{2}}{10000} + \frac{8767 \sin{\left(x \right)}}{10000}\right)\, dx$$$.
Lösning
Integrera termvis:
$${\color{red}{\int{\left(- \frac{8767 x^{2}}{10000} + \frac{8767 \sin{\left(x \right)}}{10000}\right)d x}}} = {\color{red}{\left(- \int{\frac{8767 x^{2}}{10000} d x} + \int{\frac{8767 \sin{\left(x \right)}}{10000} d x}\right)}}$$
Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{8767}{10000}$$$ och $$$f{\left(x \right)} = x^{2}$$$:
$$\int{\frac{8767 \sin{\left(x \right)}}{10000} d x} - {\color{red}{\int{\frac{8767 x^{2}}{10000} d x}}} = \int{\frac{8767 \sin{\left(x \right)}}{10000} d x} - {\color{red}{\left(\frac{8767 \int{x^{2} d x}}{10000}\right)}}$$
Tillämpa potensregeln $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ med $$$n=2$$$:
$$\int{\frac{8767 \sin{\left(x \right)}}{10000} d x} - \frac{8767 {\color{red}{\int{x^{2} d x}}}}{10000}=\int{\frac{8767 \sin{\left(x \right)}}{10000} d x} - \frac{8767 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}}{10000}=\int{\frac{8767 \sin{\left(x \right)}}{10000} d x} - \frac{8767 {\color{red}{\left(\frac{x^{3}}{3}\right)}}}{10000}$$
Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{8767}{10000}$$$ och $$$f{\left(x \right)} = \sin{\left(x \right)}$$$:
$$- \frac{8767 x^{3}}{30000} + {\color{red}{\int{\frac{8767 \sin{\left(x \right)}}{10000} d x}}} = - \frac{8767 x^{3}}{30000} + {\color{red}{\left(\frac{8767 \int{\sin{\left(x \right)} d x}}{10000}\right)}}$$
Integralen av sinus är $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:
$$- \frac{8767 x^{3}}{30000} + \frac{8767 {\color{red}{\int{\sin{\left(x \right)} d x}}}}{10000} = - \frac{8767 x^{3}}{30000} + \frac{8767 {\color{red}{\left(- \cos{\left(x \right)}\right)}}}{10000}$$
Alltså,
$$\int{\left(- \frac{8767 x^{2}}{10000} + \frac{8767 \sin{\left(x \right)}}{10000}\right)d x} = - \frac{8767 x^{3}}{30000} - \frac{8767 \cos{\left(x \right)}}{10000}$$
Förenkla:
$$\int{\left(- \frac{8767 x^{2}}{10000} + \frac{8767 \sin{\left(x \right)}}{10000}\right)d x} = - \frac{8767 \left(x^{3} + 3 \cos{\left(x \right)}\right)}{30000}$$
Lägg till integrationskonstanten:
$$\int{\left(- \frac{8767 x^{2}}{10000} + \frac{8767 \sin{\left(x \right)}}{10000}\right)d x} = - \frac{8767 \left(x^{3} + 3 \cos{\left(x \right)}\right)}{30000}+C$$
Svar
$$$\int \left(- \frac{8767 x^{2}}{10000} + \frac{8767 \sin{\left(x \right)}}{10000}\right)\, dx = - \frac{8767 \left(x^{3} + 3 \cos{\left(x \right)}\right)}{30000} + C$$$A