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人気のある微分積分 >

integral of (6+x^5)^25x^4

解

\int (6 + x^{5})^{2} 5 x^{4} d x

解

36 x^{5} + 6 x^{10} + \frac{x ^{15}}{3} + C

解答ステップ

\int (6 + x^{5})^{2} \cdot 5 x^{4} d x

定数を除く:

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

= 5 \cdot \int (6 + x^{5})^{2} x^{4} d x

u置換積分法を適用する

= 5 \cdot \int \frac{( 6 + u ) ^{2}}{5} 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}{5} \cdot \int (6 + u)^{2} d u

拡張

(6 + u)^{2} : 36 + 12 u + u^{2}

= 5 \cdot \frac{1}{5} \cdot \int 36 + 12 u + u^{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

= 5 \cdot \frac{1}{5} (\int 36 d u + \int 12 u d u + \int u^{2} d u)

\int 36 d u = 36 u

\int 12 u d u = 6 u^{2}

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

= 5 \cdot \frac{1}{5} (36 u + 6 u^{2} + \frac{u ^{3}}{3})

代用を戻す

u = x^{5}

= 5 \cdot \frac{1}{5} (36 x^{5} + 6 (x^{5})^{2} + \frac{( x ^{5} ) ^{3}}{3})

簡素化

5 \cdot \frac{1}{5} (36 x^{5} + 6 (x^{5})^{2} + \frac{( x ^{5} ) ^{3}}{3}) : 36 x^{5} + 6 x^{10} + \frac{x ^{15}}{3}

= 36 x^{5} + 6 x^{10} + \frac{x ^{15}}{3}

定数を解答に追加する

= 36 x^{5} + 6 x^{10} + \frac{x ^{15}}{3} + C

グラフ

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