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人気のある微分積分 >

integral of tan^7(x)sec^9(x)

解

\int tan^{7} (x) sec^{9} (x) d x

解

\frac{sec ^{15} ( x )}{15} - \frac{3 sec ^{13} ( x )}{13} + \frac{3 sec ^{11} ( x )}{11} - \frac{sec ^{9} ( x )}{9} + C

解答ステップ

\int tan^{7} (x) sec^{9} (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))^{3} tan (x) sec^{9} (x) d x

u置換積分法を適用する

= \int u^{8} (u^{2} - 1)^{3} d u

拡張

u^{8} (u^{2} - 1)^{3} : u^{14} - 3 u^{12} + 3 u^{10} - u^{8}

= \int u^{14} - 3 u^{12} + 3 u^{10} - u^{8} 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^{14} d u - \int 3 u^{12} d u + \int 3 u^{10} d u - \int u^{8} d u

\int u^{14} d u = \frac{u ^{15}}{15}

\int 3 u^{12} d u = \frac{3 u ^{13}}{13}

\int 3 u^{10} d u = \frac{3 u ^{11}}{11}

\int u^{8} d u = \frac{u ^{9}}{9}

= \frac{u ^{15}}{15} - \frac{3 u ^{13}}{13} + \frac{3 u ^{11}}{11} - \frac{u ^{9}}{9}

代用を戻す

u = sec (x)

= \frac{sec ^{15} ( x )}{15} - \frac{3 sec ^{13} ( x )}{13} + \frac{3 sec ^{11} ( x )}{11} - \frac{sec ^{9} ( x )}{9}

定数を解答に追加する

= \frac{sec ^{15} ( x )}{15} - \frac{3 sec ^{13} ( x )}{13} + \frac{3 sec ^{11} ( x )}{11} - \frac{sec ^{9} ( x )}{9} + C

グラフ

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