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人気のある微分積分 >

integral of 2sec^6(10x)

解

\int 2 sec^{6} (10 x) d x

解

\frac{1}{5} (tan (10 x) + \frac{2}{3} tan^{3} (10 x) + \frac{1}{5} tan^{5} (10 x)) + C

解答ステップ

\int 2 sec^{6} (10 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 sec^{6} (10 x) d x

u置換積分法を適用する

= 2 \cdot \int sec^{6} (u) \frac{1}{10} d u

定数を除く:

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

= 2 \cdot \frac{1}{10} \cdot \int sec^{6} (u) d u

三角関数の公式を使用して書き換える

= 2 \cdot \frac{1}{10} \cdot \int sec^{2} (u) (1 + tan^{2} (u))^{2} d u

u置換積分法を適用する

= 2 \cdot \frac{1}{10} \cdot \int (1 + v^{2})^{2} d v

拡張

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

= 2 \cdot \frac{1}{10} \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

= 2 \cdot \frac{1}{10} (\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}

= 2 \cdot \frac{1}{10} (v + \frac{2 v ^{3}}{3} + \frac{v ^{5}}{5})

代用を戻す

= 2 \cdot \frac{1}{10} (tan (10 x) + \frac{2 tan ^{3} ( 10 x )}{3} + \frac{tan ^{5} ( 10 x )}{5})

簡素化

2 \cdot \frac{1}{10} (tan (10 x) + \frac{2 tan ^{3} ( 10 x )}{3} + \frac{tan ^{5} ( 10 x )}{5}) : \frac{1}{5} (tan (10 x) + \frac{2}{3} tan^{3} (10 x) + \frac{1}{5} tan^{5} (10 x))

= \frac{1}{5} (tan (10 x) + \frac{2}{3} tan^{3} (10 x) + \frac{1}{5} tan^{5} (10 x))

定数を解答に追加する

= \frac{1}{5} (tan (10 x) + \frac{2}{3} tan^{3} (10 x) + \frac{1}{5} tan^{5} (10 x)) + C

グラフ

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