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人気のある微分積分 >

integral of 8sqrt(tan(x))sec^4(x)

解

\int 8 tan (x) sec^{4} (x) d x

解

8 (\frac{2}{3} tan^{\frac{3}{2}} (x) + \frac{2}{7} tan^{\frac{7}{2}} (x)) + C

解答ステップ

\int 8 tan (x) sec^{4} (x) d x

定数を除く:

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

= 8 \cdot \int tan (x) sec^{4} (x) d x

三角関数の公式を使用して書き換える

= 8 \cdot \int (1 + tan^{2} (x)) sec^{2} (x) tan (x) d x

u置換積分法を適用する

= 8 \cdot \int u (1 + u^{2}) d u

拡張

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

= 8 \cdot \int u + u^{\frac{5}{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

= 8 (\int u d u + \int u^{\frac{5}{2}} d u)

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

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

= 8 (\frac{2}{3} u^{\frac{3}{2}} + \frac{2}{7} u^{\frac{7}{2}})

代用を戻す

u = tan (x)

= 8 (\frac{2}{3} tan^{\frac{3}{2}} (x) + \frac{2}{7} tan^{\frac{7}{2}} (x))

定数を解答に追加する

= 8 (\frac{2}{3} tan^{\frac{3}{2}} (x) + \frac{2}{7} tan^{\frac{7}{2}} (x)) + C

グラフ

人気の例

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