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人気のある微分積分 >

integral of sec^{3/2}(x)tan^3(x)

解

\int sec^{\frac{3}{2}} (x) tan^{3} (x) d x

解

\frac{2}{7} sec^{\frac{7}{2}} (x) - \frac{2}{3} sec^{\frac{3}{2}} (x) + C

解答ステップ

\int sec^{\frac{3}{2}} (x) tan^{3} (x) d x

sec^{\frac{3}{2}} (x) = sec (x) sec (x)

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

簡素化

sec (x) sec^{0.5} (x) : sec^{1.5} (x)

= \int sec^{1.5} (x) tan^{3} (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)) tan (x) sec^{\frac{3}{2}} (x) d x

u置換積分法を適用する

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

拡張

u^{\frac{1}{2}} (u^{2} - 1) : u^{\frac{5}{2}} - u^{\frac{1}{2}}

= \int u^{\frac{5}{2}} - u^{\frac{1}{2}} 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^{\frac{5}{2}} d u - \int u^{\frac{1}{2}} d u

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

\int u^{\frac{1}{2}} d u = \frac{2}{3} u^{\frac{3}{2}}

= \frac{2}{7} u^{\frac{7}{2}} - \frac{2}{3} u^{\frac{3}{2}}

代用を戻す

u = sec (x)

= \frac{2}{7} sec^{\frac{7}{2}} (x) - \frac{2}{3} sec^{\frac{3}{2}} (x)

定数を解答に追加する

= \frac{2}{7} sec^{\frac{7}{2}} (x) - \frac{2}{3} sec^{\frac{3}{2}} (x) + C

グラフ

人気の例

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