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人気のある微分積分 >

integral of 2tan^4(x)sec^6(x)

解

\int 2 tan^{4} (x) sec^{6} (x) d x

解

2 (\frac{tan ^{5} ( x )}{5} + \frac{2 tan ^{7} ( x )}{7} + \frac{tan ^{9} ( x )}{9}) + C

解答ステップ

\int 2 tan^{4} (x) sec^{6} (x) d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int tan^{4} (x) sec^{6} (x) d x

三角関数の公式を使用して書き換える

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

u置換積分法を適用する

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

拡張

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

= 2 \cdot \int u^{4} + 2 u^{6} + 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

= 2 (\int u^{4} d u + \int 2 u^{6} d u + \int u^{8} d u)

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

\int 2 u^{6} d u = \frac{2 u ^{7}}{7}

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

= 2 (\frac{u ^{5}}{5} + \frac{2 u ^{7}}{7} + \frac{u ^{9}}{9})

代用を戻す

u = tan (x)

= 2 (\frac{tan ^{5} ( x )}{5} + \frac{2 tan ^{7} ( x )}{7} + \frac{tan ^{9} ( x )}{9})

定数を解答に追加する

= 2 (\frac{tan ^{5} ( x )}{5} + \frac{2 tan ^{7} ( x )}{7} + \frac{tan ^{9} ( x )}{9}) + C

グラフ

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