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人気のある微分積分 >

integral of sec^6(4x)

解

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

解

\frac{1}{4} (tan (4 x) + \frac{2 tan ^{3} ( 4 x )}{3} + \frac{tan ^{5} ( 4 x )}{5}) + C

解答ステップ

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

u置換積分法を適用する

= \int sec^{6} (u) \frac{1}{4} d u

定数を除く:

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

= \frac{1}{4} \cdot \int sec^{6} (u) d u

三角関数の公式を使用して書き換える

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

u置換積分法を適用する

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

拡張

(1 + v^{2})^{2} : 1 + 2 v^{2} + v^{4}

= \frac{1}{4} \cdot \int 1 + 2 v^{2} + v^{4} d v

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

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

\int 1 d v = v

\int 2 v^{2} d v = \frac{2 v ^{3}}{3}

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

= \frac{1}{4} (v + \frac{2 v ^{3}}{3} + \frac{v ^{5}}{5})

代用を戻す

= \frac{1}{4} (tan (4 x) + \frac{2 tan ^{3} ( 4 x )}{3} + \frac{tan ^{5} ( 4 x )}{5})

定数を解答に追加する

= \frac{1}{4} (tan (4 x) + \frac{2 tan ^{3} ( 4 x )}{3} + \frac{tan ^{5} ( 4 x )}{5}) + C

グラフ

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