Integralen av $$$\frac{- \sin^{2}{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(x \right)}}$$$
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Din inmatning
Bestäm $$$\int \frac{- \sin^{2}{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx$$$.
Lösning
Expand the expression:
$${\color{red}{\int{\frac{- \sin^{2}{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(x \right)}} d x}}} = {\color{red}{\int{\left(- \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right)d x}}}$$
Integrera termvis:
$${\color{red}{\int{\left(- \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right)d x}}} = {\color{red}{\left(\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x} - \int{\sin{\left(x \right)} d x}\right)}}$$
Integralen av sinus är $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:
$$\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x} - {\color{red}{\int{\sin{\left(x \right)} d x}}} = \int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x} - {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$
Låt $$$u=\sin{\left(x \right)}$$$ vara.
Då $$$du=\left(\sin{\left(x \right)}\right)^{\prime }dx = \cos{\left(x \right)} dx$$$ (stegen kan ses »), och vi har att $$$\cos{\left(x \right)} dx = du$$$.
Alltså,
$$\cos{\left(x \right)} + {\color{red}{\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x}}} = \cos{\left(x \right)} + {\color{red}{\int{\frac{1}{u} d u}}}$$
Integralen av $$$\frac{1}{u}$$$ är $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:
$$\cos{\left(x \right)} + {\color{red}{\int{\frac{1}{u} d u}}} = \cos{\left(x \right)} + {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$
Kom ihåg att $$$u=\sin{\left(x \right)}$$$:
$$\ln{\left(\left|{{\color{red}{u}}}\right| \right)} + \cos{\left(x \right)} = \ln{\left(\left|{{\color{red}{\sin{\left(x \right)}}}}\right| \right)} + \cos{\left(x \right)}$$
Alltså,
$$\int{\frac{- \sin^{2}{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(x \right)}} d x} = \ln{\left(\left|{\sin{\left(x \right)}}\right| \right)} + \cos{\left(x \right)}$$
Lägg till integrationskonstanten:
$$\int{\frac{- \sin^{2}{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(x \right)}} d x} = \ln{\left(\left|{\sin{\left(x \right)}}\right| \right)} + \cos{\left(x \right)}+C$$
Svar
$$$\int \frac{- \sin^{2}{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx = \left(\ln\left(\left|{\sin{\left(x \right)}}\right|\right) + \cos{\left(x \right)}\right) + C$$$A