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人気のある微分積分 >

integral of ((x^3))/(1+x^2)

解

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

解

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

解答ステップ

\int \frac{x ^{3}}{1 + x ^{2}} d x

u置換積分法を適用する

= \int \frac{u - 1}{2 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 \int \frac{u - 1}{u} d u

拡張

\frac{u - 1}{u} : 1 - \frac{1}{u}

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

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{1}{2} (u - ln ∣ u ∣)

代用を戻す

u = 1 + x^{2}

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

定数を解答に追加する

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

グラフ

人気の例

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\frac{\partial}{\partial x} (2 x^{3} y + 4)

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x \to 3 + lim (\frac{1}{x - 3})

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y^'+ytan(t)=(tan(t))/(cos(t))

y^{^{'}} + y tan (t) = \frac{tan ( t )}{cos ( t )}

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\int x^{3} I n (x) d x