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人気のある微分積分 >

integral of (x^3)/(\sqrt[3]{1-4x^2)}

解

\int \frac{x ^{3}}{3 1 - 4 x ^{2}} d x

解

\frac{1}{32} (\frac{3}{5} (1 - 4 x^{2})^{\frac{5}{3}} - \frac{3}{2} (1 - 4 x^{2})^{\frac{2}{3}}) + C

解答ステップ

\int \frac{x ^{3}}{3 1 - 4 x ^{2}} d x

u置換積分法を適用する

= \int \frac{u - 1}{32 3 u} d u

定数を除く:

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

= \frac{1}{32} \cdot \int \frac{u - 1}{3 u} d u

拡張

\frac{u - 1}{3 u} : u^{\frac{2}{3}} - \frac{1}{3 u}

= \frac{1}{32} \cdot \int u^{\frac{2}{3}} - \frac{1}{3 u} 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

= \frac{1}{32} (\int u^{\frac{2}{3}} d u - \int \frac{1}{3 u} d u)

\int u^{\frac{2}{3}} d u = \frac{3}{5} u^{\frac{5}{3}}

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

= \frac{1}{32} (\frac{3}{5} u^{\frac{5}{3}} - \frac{3}{2} u^{\frac{2}{3}})

代用を戻す

u = 1 - 4 x^{2}

= \frac{1}{32} (\frac{3}{5} (1 - 4 x^{2})^{\frac{5}{3}} - \frac{3}{2} (1 - 4 x^{2})^{\frac{2}{3}})

定数を解答に追加する

= \frac{1}{32} (\frac{3}{5} (1 - 4 x^{2})^{\frac{5}{3}} - \frac{3}{2} (1 - 4 x^{2})^{\frac{2}{3}}) + C

グラフ

人気の例

integral of 1/(xsqrt(4x+81))

\int \frac{1}{x 4 x + 81} d x

integral of ((5x-3)^3)/(x^3)

\int \frac{( 5 x - 3 ) ^{3}}{x ^{3}} d x

integral of-2pisin^2(2x)

\int - 2 π sin^{2} (2 x) d x

integral of (3(1+t^{1/2})^2)/(2t^{1/2)}

\int \frac{3 ( 1 + t ^{\frac{1}{2}} ) ^{2}}{2 t ^{\frac{1}{2}}} d t

integral of (1+sin^2(x))(1+cos^2(x))

\int (1 + sin^{2} (x)) (1 + cos^{2} (x)) d x