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人気のある微分積分 >

integral of 5x(2+3x^2)^8

解

\int 5 x (2 + 3 x^{2})^{8} d x

解

\frac{5}{54} (2 + 3 x^{2})^{9} + C

解答ステップ

\int 5 x (2 + 3 x^{2})^{8} d x

定数を除く:

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

= 5 \cdot \int x (2 + 3 x^{2})^{8} d x

u置換積分法を適用する

= 5 \cdot \int \frac{u ^{8}}{6} d u

定数を除く:

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

= 5 \cdot \frac{1}{6} \cdot \int u^{8} d u

べき乗則を適用する

= 5 \cdot \frac{1}{6} \cdot \frac{u ^{9}}{9}

代用を戻す

u = 2 + 3 x^{2}

= 5 \cdot \frac{1}{6} \cdot \frac{( 2 + 3 x ^{2} ) ^{9}}{9}

簡素化

5 \cdot \frac{1}{6} \cdot \frac{( 2 + 3 x ^{2} ) ^{9}}{9} : \frac{5}{54} (2 + 3 x^{2})^{9}

= \frac{5}{54} (2 + 3 x^{2})^{9}

定数を解答に追加する

= \frac{5}{54} (2 + 3 x^{2})^{9} + C

グラフ

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