Ολοκλήρωμα του $$$\cos^{11}{\left(x \right)}$$$

Βρείτε $$$\int \cos^{11}{\left(x \right)}\, dx$$$.

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 $$$\alpha=x$$$:

$${\color{red}{\int{\cos^{11}{\left(x \right)} d x}}} = {\color{red}{\int{\left(1 - \sin^{2}{\left(x \right)}\right)^{5} \cos{\left(x \right)} d x}}}$$

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

Το ολοκλήρωμα μπορεί να επαναγραφεί ως

$${\color{red}{\int{\left(1 - \sin^{2}{\left(x \right)}\right)^{5} \cos{\left(x \right)} d x}}} = {\color{red}{\int{\left(1 - u^{2}\right)^{5} d u}}}$$

Expand the expression:

$${\color{red}{\int{\left(1 - u^{2}\right)^{5} d u}}} = {\color{red}{\int{\left(- u^{10} + 5 u^{8} - 10 u^{6} + 10 u^{4} - 5 u^{2} + 1\right)d u}}}$$

Ολοκληρώστε όρο προς όρο:

$${\color{red}{\int{\left(- u^{10} + 5 u^{8} - 10 u^{6} + 10 u^{4} - 5 u^{2} + 1\right)d u}}} = {\color{red}{\left(\int{1 d u} - \int{5 u^{2} d u} + \int{10 u^{4} d u} - \int{10 u^{6} d u} + \int{5 u^{8} d u} - \int{u^{10} d u}\right)}}$$

Εφαρμόστε τον κανόνα της σταθεράς $$$\int c\, du = c u$$$ με $$$c=1$$$:

$$- \int{5 u^{2} d u} + \int{10 u^{4} d u} - \int{10 u^{6} d u} + \int{5 u^{8} d u} - \int{u^{10} d u} + {\color{red}{\int{1 d u}}} = - \int{5 u^{2} d u} + \int{10 u^{4} d u} - \int{10 u^{6} d u} + \int{5 u^{8} d u} - \int{u^{10} d u} + {\color{red}{u}}$$

Εφαρμόστε τον κανόνα δύναμης $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ με $$$n=10$$$:

$$u - \int{5 u^{2} d u} + \int{10 u^{4} d u} - \int{10 u^{6} d u} + \int{5 u^{8} d u} - {\color{red}{\int{u^{10} d u}}}=u - \int{5 u^{2} d u} + \int{10 u^{4} d u} - \int{10 u^{6} d u} + \int{5 u^{8} d u} - {\color{red}{\frac{u^{1 + 10}}{1 + 10}}}=u - \int{5 u^{2} d u} + \int{10 u^{4} d u} - \int{10 u^{6} d u} + \int{5 u^{8} d u} - {\color{red}{\left(\frac{u^{11}}{11}\right)}}$$

Εφαρμόστε τον κανόνα του σταθερού πολλαπλασίου $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ με $$$c=10$$$ και $$$f{\left(u \right)} = u^{6}$$$:

$$- \frac{u^{11}}{11} + u - \int{5 u^{2} d u} + \int{10 u^{4} d u} + \int{5 u^{8} d u} - {\color{red}{\int{10 u^{6} d u}}} = - \frac{u^{11}}{11} + u - \int{5 u^{2} d u} + \int{10 u^{4} d u} + \int{5 u^{8} d u} - {\color{red}{\left(10 \int{u^{6} d u}\right)}}$$

Εφαρμόστε τον κανόνα δύναμης $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ με $$$n=6$$$:

$$- \frac{u^{11}}{11} + u - \int{5 u^{2} d u} + \int{10 u^{4} d u} + \int{5 u^{8} d u} - 10 {\color{red}{\int{u^{6} d u}}}=- \frac{u^{11}}{11} + u - \int{5 u^{2} d u} + \int{10 u^{4} d u} + \int{5 u^{8} d u} - 10 {\color{red}{\frac{u^{1 + 6}}{1 + 6}}}=- \frac{u^{11}}{11} + u - \int{5 u^{2} d u} + \int{10 u^{4} d u} + \int{5 u^{8} d u} - 10 {\color{red}{\left(\frac{u^{7}}{7}\right)}}$$

Εφαρμόστε τον κανόνα του σταθερού πολλαπλασίου $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ με $$$c=5$$$ και $$$f{\left(u \right)} = u^{2}$$$:

$$- \frac{u^{11}}{11} - \frac{10 u^{7}}{7} + u + \int{10 u^{4} d u} + \int{5 u^{8} d u} - {\color{red}{\int{5 u^{2} d u}}} = - \frac{u^{11}}{11} - \frac{10 u^{7}}{7} + u + \int{10 u^{4} d u} + \int{5 u^{8} d u} - {\color{red}{\left(5 \int{u^{2} d u}\right)}}$$

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

$$- \frac{u^{11}}{11} - \frac{10 u^{7}}{7} + u + \int{10 u^{4} d u} + \int{5 u^{8} d u} - 5 {\color{red}{\int{u^{2} d u}}}=- \frac{u^{11}}{11} - \frac{10 u^{7}}{7} + u + \int{10 u^{4} d u} + \int{5 u^{8} d u} - 5 {\color{red}{\frac{u^{1 + 2}}{1 + 2}}}=- \frac{u^{11}}{11} - \frac{10 u^{7}}{7} + u + \int{10 u^{4} d u} + \int{5 u^{8} d u} - 5 {\color{red}{\left(\frac{u^{3}}{3}\right)}}$$

Εφαρμόστε τον κανόνα του σταθερού πολλαπλασίου $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ με $$$c=5$$$ και $$$f{\left(u \right)} = u^{8}$$$:

$$- \frac{u^{11}}{11} - \frac{10 u^{7}}{7} - \frac{5 u^{3}}{3} + u + \int{10 u^{4} d u} + {\color{red}{\int{5 u^{8} d u}}} = - \frac{u^{11}}{11} - \frac{10 u^{7}}{7} - \frac{5 u^{3}}{3} + u + \int{10 u^{4} d u} + {\color{red}{\left(5 \int{u^{8} d u}\right)}}$$

Εφαρμόστε τον κανόνα δύναμης $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ με $$$n=8$$$:

$$- \frac{u^{11}}{11} - \frac{10 u^{7}}{7} - \frac{5 u^{3}}{3} + u + \int{10 u^{4} d u} + 5 {\color{red}{\int{u^{8} d u}}}=- \frac{u^{11}}{11} - \frac{10 u^{7}}{7} - \frac{5 u^{3}}{3} + u + \int{10 u^{4} d u} + 5 {\color{red}{\frac{u^{1 + 8}}{1 + 8}}}=- \frac{u^{11}}{11} - \frac{10 u^{7}}{7} - \frac{5 u^{3}}{3} + u + \int{10 u^{4} d u} + 5 {\color{red}{\left(\frac{u^{9}}{9}\right)}}$$

Εφαρμόστε τον κανόνα του σταθερού πολλαπλασίου $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ με $$$c=10$$$ και $$$f{\left(u \right)} = u^{4}$$$:

$$- \frac{u^{11}}{11} + \frac{5 u^{9}}{9} - \frac{10 u^{7}}{7} - \frac{5 u^{3}}{3} + u + {\color{red}{\int{10 u^{4} d u}}} = - \frac{u^{11}}{11} + \frac{5 u^{9}}{9} - \frac{10 u^{7}}{7} - \frac{5 u^{3}}{3} + u + {\color{red}{\left(10 \int{u^{4} d u}\right)}}$$

Εφαρμόστε τον κανόνα δύναμης $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ με $$$n=4$$$:

$$- \frac{u^{11}}{11} + \frac{5 u^{9}}{9} - \frac{10 u^{7}}{7} - \frac{5 u^{3}}{3} + u + 10 {\color{red}{\int{u^{4} d u}}}=- \frac{u^{11}}{11} + \frac{5 u^{9}}{9} - \frac{10 u^{7}}{7} - \frac{5 u^{3}}{3} + u + 10 {\color{red}{\frac{u^{1 + 4}}{1 + 4}}}=- \frac{u^{11}}{11} + \frac{5 u^{9}}{9} - \frac{10 u^{7}}{7} - \frac{5 u^{3}}{3} + u + 10 {\color{red}{\left(\frac{u^{5}}{5}\right)}}$$

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

$${\color{red}{u}} - \frac{5 {\color{red}{u}}^{3}}{3} + 2 {\color{red}{u}}^{5} - \frac{10 {\color{red}{u}}^{7}}{7} + \frac{5 {\color{red}{u}}^{9}}{9} - \frac{{\color{red}{u}}^{11}}{11} = {\color{red}{\sin{\left(x \right)}}} - \frac{5 {\color{red}{\sin{\left(x \right)}}}^{3}}{3} + 2 {\color{red}{\sin{\left(x \right)}}}^{5} - \frac{10 {\color{red}{\sin{\left(x \right)}}}^{7}}{7} + \frac{5 {\color{red}{\sin{\left(x \right)}}}^{9}}{9} - \frac{{\color{red}{\sin{\left(x \right)}}}^{11}}{11}$$

Επομένως,

$$\int{\cos^{11}{\left(x \right)} d x} = - \frac{\sin^{11}{\left(x \right)}}{11} + \frac{5 \sin^{9}{\left(x \right)}}{9} - \frac{10 \sin^{7}{\left(x \right)}}{7} + 2 \sin^{5}{\left(x \right)} - \frac{5 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$$

Απλοποιήστε:

$$\int{\cos^{11}{\left(x \right)} d x} = \frac{\left(- 63 \sin^{10}{\left(x \right)} + 385 \sin^{8}{\left(x \right)} - 990 \sin^{6}{\left(x \right)} + 1386 \sin^{4}{\left(x \right)} - 1155 \sin^{2}{\left(x \right)} + 693\right) \sin{\left(x \right)}}{693}$$

Προσθέστε τη σταθερά ολοκλήρωσης:

$$\int{\cos^{11}{\left(x \right)} d x} = \frac{\left(- 63 \sin^{10}{\left(x \right)} + 385 \sin^{8}{\left(x \right)} - 990 \sin^{6}{\left(x \right)} + 1386 \sin^{4}{\left(x \right)} - 1155 \sin^{2}{\left(x \right)} + 693\right) \sin{\left(x \right)}}{693}+C$$

$$$\int \cos^{11}{\left(x \right)}\, dx = \frac{\left(- 63 \sin^{10}{\left(x \right)} + 385 \sin^{8}{\left(x \right)} - 990 \sin^{6}{\left(x \right)} + 1386 \sin^{4}{\left(x \right)} - 1155 \sin^{2}{\left(x \right)} + 693\right) \sin{\left(x \right)}}{693} + C$$$A