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