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人気のある微分積分 >

integral of sec^{10}(x)tan(x)

解

\int sec^{10} (x) tan (x) d x

解

\frac{tan ^{2} ( x )}{2} + tan^{4} (x) + tan^{6} (x) + \frac{tan ^{8} ( x )}{2} + \frac{tan ^{10} ( x )}{10} + C

解答ステップ

\int sec^{10} (x) tan (x) d x

三角関数の公式を使用して書き換える

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

u置換積分法を適用する

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

拡張

u (1 + u^{2})^{4} : u + 4 u^{3} + 6 u^{5} + 4 u^{7} + u^{9}

= \int u + 4 u^{3} + 6 u^{5} + 4 u^{7} + u^{9} 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 d u + \int 4 u^{3} d u + \int 6 u^{5} d u + \int 4 u^{7} d u + \int u^{9} d u

\int u d u = \frac{u ^{2}}{2}

\int 4 u^{3} d u = u^{4}

\int 6 u^{5} d u = u^{6}

\int 4 u^{7} d u = \frac{u ^{8}}{2}

\int u^{9} d u = \frac{u ^{10}}{10}

= \frac{u ^{2}}{2} + u^{4} + u^{6} + \frac{u ^{8}}{2} + \frac{u ^{10}}{10}

代用を戻す

u = tan (x)

= \frac{tan ^{2} ( x )}{2} + tan^{4} (x) + tan^{6} (x) + \frac{tan ^{8} ( x )}{2} + \frac{tan ^{10} ( x )}{10}

定数を解答に追加する

= \frac{tan ^{2} ( x )}{2} + tan^{4} (x) + tan^{6} (x) + \frac{tan ^{8} ( x )}{2} + \frac{tan ^{10} ( x )}{10} + C

グラフ

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