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人気のある微分積分 >

integral of sin^3(3x)sqrt(cos(3x))

解

\int sin^{3} (3 x) cos (3 x) d x

解

- \frac{1}{3} (\frac{2}{3} cos^{\frac{3}{2}} (3 x) - \frac{2}{7} cos^{\frac{7}{2}} (3 x)) + C

解答ステップ

\int sin^{3} (3 x) cos (3 x) d x

u置換積分法を適用する

= \int sin^{3} (u) cos (u) \frac{1}{3} d u

定数を除く:

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

= \frac{1}{3} \cdot \int sin^{3} (u) cos (u) d u

u置換積分法を適用する

= \frac{1}{3} \cdot \int - v (1 - v^{2}) d v

定数を除く:

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

= \frac{1}{3} (- \int v (1 - v^{2}) d v)

拡張

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

= \frac{1}{3} (- \int v - v^{\frac{5}{2}} 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}{3} (- (\int v d v - \int v^{\frac{5}{2}} d v))

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

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

= \frac{1}{3} (- (\frac{2}{3} v^{\frac{3}{2}} - \frac{2}{7} v^{\frac{7}{2}}))

代用を戻す

= \frac{1}{3} (- (\frac{2}{3} cos^{\frac{3}{2}} (3 x) - \frac{2}{7} cos^{\frac{7}{2}} (3 x)))

簡素化

= - \frac{1}{3} (\frac{2}{3} cos^{\frac{3}{2}} (3 x) - \frac{2}{7} cos^{\frac{7}{2}} (3 x))

定数を解答に追加する

= - \frac{1}{3} (\frac{2}{3} cos^{\frac{3}{2}} (3 x) - \frac{2}{7} cos^{\frac{7}{2}} (3 x)) + C

グラフ

人気の例

integral of sec^2(x)csc^2(x)

\int sec^{2} (x) csc^{2} (x) d x

integral from 1/2 to 7 of 10xln(2x)

\int_{\frac{1}{2}}^{7} 10 x ln (2 x) d x

integral from 0 to 4 of 3e^{-x^2}

\int_{0}^{4} 3 e^{- x^{2}} d x

derivative f(x)=9

d er i v a t i v e f (x) = 9

integral of cot(m)xcsc^2(m)x

\int cot (m) x csc^{2} (m) x d x