integral of sec^{11}(x)

解

\int sec^{11} (x) d x

\frac{1}{10} sec^{10} (x) sin (x) + \frac{3}{1280} (48 sec^{8} (x) sin (x) + 7 (8 sec^{6} (x) sin (x) + 5 (2 sec^{4} (x) sin (x) + 3 (sec (x) tan (x) + ln ∣ tan (x) + sec (x) ∣)))) + C

解答ステップ

\int sec^{11} (x) d x

積分換算を適用する

= \frac{sec ^{10} ( x ) sin ( x )}{10} + \frac{9}{10} \cdot \int sec^{9} (x) d x

\int sec^{9} (x) d x = \frac{sec ^{8} ( x ) sin ( x )}{8} + \frac{7}{8} (\frac{1}{6} sec^{6} (x) sin (x) + \frac{5}{48} (2 sec^{4} (x) sin (x) + 3 (sec (x) tan (x) + ln ∣ tan (x) + sec (x) ∣)))

= \frac{sec ^{10} ( x ) sin ( x )}{10} + \frac{9}{10} (\frac{sec ^{8} ( x ) sin ( x )}{8} + \frac{7}{8} (\frac{1}{6} sec^{6} (x) sin (x) + \frac{5}{48} (2 sec^{4} (x) sin (x) + 3 (sec (x) tan (x) + ln ∣ tan (x) + sec (x) ∣))))

簡素化

\frac{sec ^{10} ( x ) sin ( x )}{10} + \frac{9}{10} (\frac{sec ^{8} ( x ) sin ( x )}{8} + \frac{7}{8} (\frac{1}{6} sec^{6} (x) sin (x) + \frac{5}{48} (2 sec^{4} (x) sin (x) + 3 (sec (x) tan (x) + ln ∣ tan (x) + sec (x) ∣)))) : \frac{1}{10} sec^{10} (x) sin (x) + \frac{3}{1280} (48 sec^{8} (x) sin (x) + 7 (8 sec^{6} (x) sin (x) + 5 (2 sec^{4} (x) sin (x) + 3 (sec (x) tan (x) + ln ∣ tan (x) + sec (x) ∣))))

= \frac{1}{10} sec^{10} (x) sin (x) + \frac{3}{1280} (48 sec^{8} (x) sin (x) + 7 (8 sec^{6} (x) sin (x) + 5 (2 sec^{4} (x) sin (x) + 3 (sec (x) tan (x) + ln ∣ tan (x) + sec (x) ∣))))

定数を解答に追加する

= \frac{1}{10} sec^{10} (x) sin (x) + \frac{3}{1280} (48 sec^{8} (x) sin (x) + 7 (8 sec^{6} (x) sin (x) + 5 (2 sec^{4} (x) sin (x) + 3 (sec (x) tan (x) + ln ∣ tan (x) + sec (x) ∣)))) + C