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人気のある微分積分 >

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

解

\int 3 x^{2} ln (3 x - 1) d x

解

x^{2} ln (x - 1) - x ln (x - 1) + \frac{1}{3} ln (x - 1) (x - 1)^{3} - \frac{1}{9} (x - 1)^{3} - \frac{1}{2} (x - 1)^{2} - x + 1 + C

解答ステップ

\int 3 x^{2} ln (3 x - 1) 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} ln (3 x - 1) d x

簡素化

ln (3 x - 1) : \frac{1}{3} ln (x - 1)

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

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 x^{2} ln (x - 1) d x

u置換積分法を適用する

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

部分積分を適用する

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

\int \frac{1}{3} (u^{2} + 3 u + 3) d u = \frac{u ^{3}}{9} + \frac{u ^{2}}{2} + u

= 3 \cdot \frac{1}{3} (ln (u) (\frac{u ^{3}}{3} + u^{2} + u) - (\frac{u ^{3}}{9} + \frac{u ^{2}}{2} + u))

代用を戻す

u = x - 1

= 3 \cdot \frac{1}{3} (ln (x - 1) (\frac{( x - 1 ) ^{3}}{3} + (x - 1)^{2} + x - 1) - (\frac{( x - 1 ) ^{3}}{9} + \frac{( x - 1 ) ^{2}}{2} + x - 1))

簡素化

3 \cdot \frac{1}{3} (ln (x - 1) (\frac{( x - 1 ) ^{3}}{3} + (x - 1)^{2} + x - 1) - (\frac{( x - 1 ) ^{3}}{9} + \frac{( x - 1 ) ^{2}}{2} + x - 1)) : x^{2} ln (x - 1) - x ln (x - 1) + \frac{1}{3} ln (x - 1) (x - 1)^{3} - \frac{1}{9} (x - 1)^{3} - \frac{1}{2} (x - 1)^{2} - x + 1

= x^{2} ln (x - 1) - x ln (x - 1) + \frac{1}{3} ln (x - 1) (x - 1)^{3} - \frac{1}{9} (x - 1)^{3} - \frac{1}{2} (x - 1)^{2} - x + 1

定数を解答に追加する

= x^{2} ln (x - 1) - x ln (x - 1) + \frac{1}{3} ln (x - 1) (x - 1)^{3} - \frac{1}{9} (x - 1)^{3} - \frac{1}{2} (x - 1)^{2} - x + 1 + C

グラフ

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