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人気のある微分積分 >

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

解

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

解

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

解答ステップ

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

定数を除く:

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

= 3 \cdot \int x^{2} (x^{3} + 1)^{\frac{1}{2}} d x

u置換積分法を適用する

= 3 \cdot \int \frac{( u + 1 ) ^{\frac{1}{2}}}{3} d u

定数を除く:

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

= 3 \cdot \frac{1}{3} \cdot \int (u + 1)^{\frac{1}{2}} d u

部分積分を適用する

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

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

= 3 \cdot \frac{1}{3} (u (u + 1)^{\frac{1}{2}} - \frac{1}{3} (u - 2) (u + 1)^{\frac{1}{2}})

代用を戻す

u = x^{3}

= 3 \cdot \frac{1}{3} (x^{3} (x^{3} + 1)^{\frac{1}{2}} - \frac{1}{3} (x^{3} - 2) (x^{3} + 1)^{\frac{1}{2}})

簡素化

3 \cdot \frac{1}{3} (x^{3} (x^{3} + 1)^{\frac{1}{2}} - \frac{1}{3} (x^{3} - 2) (x^{3} + 1)^{\frac{1}{2}}) : \frac{2}{3} x^{3} (x^{3} + 1)^{\frac{1}{2}} + \frac{2}{3} (x^{3} + 1)^{\frac{1}{2}}

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

定数を解答に追加する

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

グラフ

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