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人気のある微分積分 >

integral of (ln(3x+1))/((6x+2))

解

\int \frac{ln ( 3 x + 1 )}{( 6 x + 2 )} d x

解

\frac{1}{12} ln^{2} (3 x + 1) + C

解答ステップ

\int \frac{ln ( 3 x + 1 )}{6 x + 2} d x

\frac{ln ( 3 x + 1 )}{6 x + 2} = \frac{1}{2} \frac{ln ( 3 x + 1 )}{3 x + 1}

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

定数を除く:

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

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

u置換積分法を適用する

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

u置換積分法を適用する

= \frac{1}{2} \cdot \frac{1}{3} \cdot \int v d v

べき乗則を適用する

= \frac{1}{2} \cdot \frac{1}{3} \cdot \frac{v ^{2}}{2}

代用を戻す

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

簡素化

\frac{1}{2} \cdot \frac{1}{3} \cdot \frac{ln ^{2} ( 3 x + 1 )}{2} : \frac{1}{12} ln^{2} (3 x + 1)

= \frac{1}{12} ln^{2} (3 x + 1)

定数を解答に追加する

= \frac{1}{12} ln^{2} (3 x + 1) + C

グラフ

人気の例

y^{^{'}} = y (3 - t y)

integral of (3x^2-x+5)/(x^4)

\int \frac{3 x ^{2} - x + 5}{x ^{4}} d x

y^{^{'}} = - \frac{x}{y}

limit as x approaches-3 of (x+2)/(x-2)

x \to - 3 lim (\frac{x + 2}{x - 2})

(\partial)/(\partial x)(cos^2(x)y)

\frac{\partial}{\partial x} (cos^{2} (x) y)