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人気のある微分積分 >

integral of (5x^2-12x)e^{2x}

解

\int (5 x^{2} - 12 x) e^{2 x} d x

解

\frac{1}{4} e^{2 x} (10 x^{2} - 34 x + 17) + C

解答ステップ

\int (5 x^{2} - 12 x) e^{2 x} d x

u置換積分法を適用する

= \int \frac{5 ln ^{2} ( u ) - 24 ln ( u )}{8} d u

定数を除く:

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

= \frac{1}{8} \cdot \int 5 ln^{2} (u) - 24 ln (u) 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

= \frac{1}{8} (\int 5 ln^{2} (u) d u - \int 24 ln (u) d u)

\int 5 ln^{2} (u) d u = 5 (u ln^{2} (u) - 2 (u ln (u) - u))

\int 24 ln (u) d u = 24 (u ln (u) - u)

= \frac{1}{8} (5 (u ln^{2} (u) - 2 (u ln (u) - u)) - 24 (u ln (u) - u))

代用を戻す

u = e^{2 x}

= \frac{1}{8} (5 (e^{2 x} ln^{2} (e^{2 x}) - 2 (e^{2 x} ln (e^{2 x}) - e^{2 x})) - 24 (e^{2 x} ln (e^{2 x}) - e^{2 x}))

簡素化

\frac{1}{8} (5 (e^{2 x} ln^{2} (e^{2 x}) - 2 (e^{2 x} ln (e^{2 x}) - e^{2 x})) - 24 (e^{2 x} ln (e^{2 x}) - e^{2 x})) : \frac{1}{4} e^{2 x} (10 x^{2} - 34 x + 17)

= \frac{1}{4} e^{2 x} (10 x^{2} - 34 x + 17)

定数を解答に追加する

= \frac{1}{4} e^{2 x} (10 x^{2} - 34 x + 17) + C

グラフ

人気の例

derivative of x/(sqrt(x^2+3))

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

derivative of (cot(x)/(sin(x)))

\frac{d}{d x} (\frac{cot ( x )}{sin ( x )})

derivative of x^3+x^2-x

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

limit as x approaches-2 of (-9)/(x^2)

x \to - 2 lim (\frac{- 9}{x ^{2}})

limit as x approaches infinity of (5)\sqrt[6666]{6}

x \to \infty lim ((5) 66666)