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人気のある微分積分 >

integral of xln(1+x^2)

解

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

解

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

解答ステップ

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

u置換積分法を適用する

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

部分積分を適用する

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

\int 1 d u = u

= \frac{1}{2} (u ln (u) - u)

代用を戻す

u = 1 + x^{2}

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

拡張

(1 + x^{2}) ln (1 + x^{2}) - (1 + x^{2}) : ln (1 + x^{2}) + x^{2} ln (1 + x^{2}) - 1 - x^{2}

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

定数を解答に追加する

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

グラフ

人気の例

(dy)/(dx)=(y/x)

\frac{d y}{d x} = (\frac{y}{x})

derivative of x^4sin(xsin(3x))

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

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integral of xsin(-4x)

\int x sin (- 4 x) d x