Ολοκλήρωμα του $$$\frac{- \sin^{2}{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(x \right)}}$$$

Βρείτε $$$\int \frac{- \sin^{2}{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx$$$.

Expand the expression:

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

Ολοκληρώστε όρο προς όρο:

$${\color{red}{\int{\left(- \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right)d x}}} = {\color{red}{\left(\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x} - \int{\sin{\left(x \right)} d x}\right)}}$$

Το ολοκλήρωμα του ημιτόνου είναι $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

$$\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x} - {\color{red}{\int{\sin{\left(x \right)} d x}}} = \int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x} - {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

Έστω $$$u=\sin{\left(x \right)}$$$.

Τότε $$$du=\left(\sin{\left(x \right)}\right)^{\prime }dx = \cos{\left(x \right)} dx$$$ (τα βήματα παρουσιάζονται »), και έχουμε ότι $$$\cos{\left(x \right)} dx = du$$$.

Το ολοκλήρωμα γίνεται

$$\cos{\left(x \right)} + {\color{red}{\int{\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} d x}}} = \cos{\left(x \right)} + {\color{red}{\int{\frac{1}{u} d u}}}$$

Το ολοκλήρωμα του $$$\frac{1}{u}$$$ είναι $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

Θυμηθείτε ότι $$$u=\sin{\left(x \right)}$$$:

$$\ln{\left(\left|{{\color{red}{u}}}\right| \right)} + \cos{\left(x \right)} = \ln{\left(\left|{{\color{red}{\sin{\left(x \right)}}}}\right| \right)} + \cos{\left(x \right)}$$

Επομένως,

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

Προσθέστε τη σταθερά ολοκλήρωσης:

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

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