Integralen av $$$\frac{x^{2}}{\left(c - x\right)^{2}}$$$ med avseende på $$$x$$$

Bestäm $$$\int \frac{x^{2}}{\left(c - x\right)^{2}}\, dx$$$.

Eftersom graden hos täljaren inte är mindre än graden hos nämnaren, utför polynomdivision.:

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

Integrera termvis:

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

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=1$$$:

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

Förenkla integranden:

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

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

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

Skriv om integrandens täljare som $$$- c + 2 x=-2\left(c - x\right)+c$$$ och dela upp bråket:

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

Integrera termvis:

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=2$$$ och $$$f{\left(x \right)} = \frac{1}{c - x}$$$:

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

Låt $$$u=c - x$$$ vara.

Då $$$du=\left(c - x\right)^{\prime }dx = - dx$$$ (stegen kan ses »), och vi har att $$$dx = - du$$$.

Alltså,

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

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=-1$$$ och $$$f{\left(u \right)} = \frac{1}{u}$$$:

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

Integralen av $$$\frac{1}{u}$$$ är $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$$c \left(\int{\frac{c}{\left(c - x\right)^{2}} d x} + 2 {\color{red}{\int{\frac{1}{u} d u}}}\right) + x = c \left(\int{\frac{c}{\left(c - x\right)^{2}} d x} + 2 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}\right) + x$$

Kom ihåg att $$$u=c - x$$$:

$$c \left(2 \ln{\left(\left|{{\color{red}{u}}}\right| \right)} + \int{\frac{c}{\left(c - x\right)^{2}} d x}\right) + x = c \left(2 \ln{\left(\left|{{\color{red}{\left(c - x\right)}}}\right| \right)} + \int{\frac{c}{\left(c - x\right)^{2}} d x}\right) + x$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=c$$$ och $$$f{\left(x \right)} = \frac{1}{\left(c - x\right)^{2}}$$$:

$$c \left(2 \ln{\left(\left|{c - x}\right| \right)} + {\color{red}{\int{\frac{c}{\left(c - x\right)^{2}} d x}}}\right) + x = c \left(2 \ln{\left(\left|{c - x}\right| \right)} + {\color{red}{c \int{\frac{1}{\left(c - x\right)^{2}} d x}}}\right) + x$$

Låt $$$u=c - x$$$ vara.

Då $$$du=\left(c - x\right)^{\prime }dx = - dx$$$ (stegen kan ses »), och vi har att $$$dx = - du$$$.

Alltså,

$$c \left(c {\color{red}{\int{\frac{1}{\left(c - x\right)^{2}} d x}}} + 2 \ln{\left(\left|{c - x}\right| \right)}\right) + x = c \left(c {\color{red}{\int{\left(- \frac{1}{u^{2}}\right)d u}}} + 2 \ln{\left(\left|{c - x}\right| \right)}\right) + x$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=-1$$$ och $$$f{\left(u \right)} = \frac{1}{u^{2}}$$$:

$$c \left(c {\color{red}{\int{\left(- \frac{1}{u^{2}}\right)d u}}} + 2 \ln{\left(\left|{c - x}\right| \right)}\right) + x = c \left(c {\color{red}{\left(- \int{\frac{1}{u^{2}} d u}\right)}} + 2 \ln{\left(\left|{c - x}\right| \right)}\right) + x$$

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

$$c \left(- c {\color{red}{\int{\frac{1}{u^{2}} d u}}} + 2 \ln{\left(\left|{c - x}\right| \right)}\right) + x=c \left(- c {\color{red}{\int{u^{-2} d u}}} + 2 \ln{\left(\left|{c - x}\right| \right)}\right) + x=c \left(- c {\color{red}{\frac{u^{-2 + 1}}{-2 + 1}}} + 2 \ln{\left(\left|{c - x}\right| \right)}\right) + x=c \left(- c {\color{red}{\left(- u^{-1}\right)}} + 2 \ln{\left(\left|{c - x}\right| \right)}\right) + x=c \left(- c {\color{red}{\left(- \frac{1}{u}\right)}} + 2 \ln{\left(\left|{c - x}\right| \right)}\right) + x$$

Kom ihåg att $$$u=c - x$$$:

$$c \left(c {\color{red}{u}}^{-1} + 2 \ln{\left(\left|{c - x}\right| \right)}\right) + x = c \left(c {\color{red}{\left(c - x\right)}}^{-1} + 2 \ln{\left(\left|{c - x}\right| \right)}\right) + x$$

Alltså,

$$\int{\frac{x^{2}}{\left(c - x\right)^{2}} d x} = c \left(\frac{c}{c - x} + 2 \ln{\left(\left|{c - x}\right| \right)}\right) + x$$

Förenkla:

$$\int{\frac{x^{2}}{\left(c - x\right)^{2}} d x} = \frac{- c \left(c - 2 \left(- c + x\right) \ln{\left(\left|{c - x}\right| \right)}\right) + x \left(- c + x\right)}{- c + x}$$

Lägg till integrationskonstanten:

$$\int{\frac{x^{2}}{\left(c - x\right)^{2}} d x} = \frac{- c \left(c - 2 \left(- c + x\right) \ln{\left(\left|{c - x}\right| \right)}\right) + x \left(- c + x\right)}{- c + x}+C$$

$$$\int \frac{x^{2}}{\left(c - x\right)^{2}}\, dx = \frac{- c \left(c - 2 \left(- c + x\right) \ln\left(\left|{c - x}\right|\right)\right) + x \left(- c + x\right)}{- c + x} + C$$$A