Ολοκλήρωμα του $$$\frac{\cos^{3}{\left(x \right)}}{\sqrt{\sin{\left(x \right)}}}$$$

Βρείτε $$$\int \frac{\cos^{3}{\left(x \right)}}{\sqrt{\sin{\left(x \right)}}}\, dx$$$.

Απομονώστε ένα συνημίτονο και εκφράστε τα υπόλοιπα σε όρους του ημιτόνου, χρησιμοποιώντας τον τύπο $$$\cos^2\left(\alpha \right)=-\sin^2\left(\alpha \right)+1$$$ με $$$\alpha=x$$$:

$${\color{red}{\int{\frac{\cos^{3}{\left(x \right)}}{\sqrt{\sin{\left(x \right)}}} d x}}} = {\color{red}{\int{\frac{\left(1 - \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)}}{\sqrt{\sin{\left(x \right)}}} d x}}}$$

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 $$$\cos{\left(x \right)} dx = du$$$.

Επομένως,

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

Expand the expression:

$${\color{red}{\int{\frac{1 - u^{2}}{\sqrt{u}} d u}}} = {\color{red}{\int{\left(- u^{\frac{3}{2}} + \frac{1}{\sqrt{u}}\right)d u}}}$$

Ολοκληρώστε όρο προς όρο:

$${\color{red}{\int{\left(- u^{\frac{3}{2}} + \frac{1}{\sqrt{u}}\right)d u}}} = {\color{red}{\left(\int{\frac{1}{\sqrt{u}} d u} - \int{u^{\frac{3}{2}} d u}\right)}}$$

Εφαρμόστε τον κανόνα δύναμης $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ με $$$n=- \frac{1}{2}$$$:

$$- \int{u^{\frac{3}{2}} d u} + {\color{red}{\int{\frac{1}{\sqrt{u}} d u}}}=- \int{u^{\frac{3}{2}} d u} + {\color{red}{\int{u^{- \frac{1}{2}} d u}}}=- \int{u^{\frac{3}{2}} d u} + {\color{red}{\frac{u^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}=- \int{u^{\frac{3}{2}} d u} + {\color{red}{\left(2 u^{\frac{1}{2}}\right)}}=- \int{u^{\frac{3}{2}} d u} + {\color{red}{\left(2 \sqrt{u}\right)}}$$

Εφαρμόστε τον κανόνα δύναμης $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ με $$$n=\frac{3}{2}$$$:

$$2 \sqrt{u} - {\color{red}{\int{u^{\frac{3}{2}} d u}}}=2 \sqrt{u} - {\color{red}{\frac{u^{1 + \frac{3}{2}}}{1 + \frac{3}{2}}}}=2 \sqrt{u} - {\color{red}{\left(\frac{2 u^{\frac{5}{2}}}{5}\right)}}$$

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

$$2 \sqrt{{\color{red}{u}}} - \frac{2 {\color{red}{u}}^{\frac{5}{2}}}{5} = 2 \sqrt{{\color{red}{\sin{\left(x \right)}}}} - \frac{2 {\color{red}{\sin{\left(x \right)}}}^{\frac{5}{2}}}{5}$$

Επομένως,

$$\int{\frac{\cos^{3}{\left(x \right)}}{\sqrt{\sin{\left(x \right)}}} d x} = - \frac{2 \sin^{\frac{5}{2}}{\left(x \right)}}{5} + 2 \sqrt{\sin{\left(x \right)}}$$

Απλοποιήστε:

$$\int{\frac{\cos^{3}{\left(x \right)}}{\sqrt{\sin{\left(x \right)}}} d x} = \frac{2 \left(5 - \sin^{2}{\left(x \right)}\right) \sqrt{\sin{\left(x \right)}}}{5}$$

Προσθέστε τη σταθερά ολοκλήρωσης:

$$\int{\frac{\cos^{3}{\left(x \right)}}{\sqrt{\sin{\left(x \right)}}} d x} = \frac{2 \left(5 - \sin^{2}{\left(x \right)}\right) \sqrt{\sin{\left(x \right)}}}{5}+C$$

$$$\int \frac{\cos^{3}{\left(x \right)}}{\sqrt{\sin{\left(x \right)}}}\, dx = \frac{2 \left(5 - \sin^{2}{\left(x \right)}\right) \sqrt{\sin{\left(x \right)}}}{5} + C$$$A