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人気のある微分積分 >

integral of tan^9(x)sec^5(x)

解

\int tan^{9} (x) sec^{5} (x) d x

解

\frac{sec ^{13} ( x )}{13} - \frac{4 sec ^{11} ( x )}{11} + \frac{2 sec ^{9} ( x )}{3} - \frac{4 sec ^{7} ( x )}{7} + \frac{sec ^{5} ( x )}{5} + C

解答ステップ

\int tan^{9} (x) sec^{5} (x) d x

三角関数の公式を使用して書き換える

= \int (- 1 + sec^{2} (x))^{4} tan (x) sec^{5} (x) d x

u置換積分法を適用する

= \int u^{4} (u^{2} - 1)^{4} d u

拡張

u^{4} (u^{2} - 1)^{4} : u^{12} - 4 u^{10} + 6 u^{8} - 4 u^{6} + u^{4}

= \int u^{12} - 4 u^{10} + 6 u^{8} - 4 u^{6} + u^{4} d u

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u^{12} d u - \int 4 u^{10} d u + \int 6 u^{8} d u - \int 4 u^{6} d u + \int u^{4} d u

\int u^{12} d u = \frac{u ^{13}}{13}

\int 4 u^{10} d u = \frac{4 u ^{11}}{11}

\int 6 u^{8} d u = \frac{2 u ^{9}}{3}

\int 4 u^{6} d u = \frac{4 u ^{7}}{7}

\int u^{4} d u = \frac{u ^{5}}{5}

= \frac{u ^{13}}{13} - \frac{4 u ^{11}}{11} + \frac{2 u ^{9}}{3} - \frac{4 u ^{7}}{7} + \frac{u ^{5}}{5}

代用を戻す

u = sec (x)

= \frac{sec ^{13} ( x )}{13} - \frac{4 sec ^{11} ( x )}{11} + \frac{2 sec ^{9} ( x )}{3} - \frac{4 sec ^{7} ( x )}{7} + \frac{sec ^{5} ( x )}{5}

定数を解答に追加する

= \frac{sec ^{13} ( x )}{13} - \frac{4 sec ^{11} ( x )}{11} + \frac{2 sec ^{9} ( x )}{3} - \frac{4 sec ^{7} ( x )}{7} + \frac{sec ^{5} ( x )}{5} + C

グラフ

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