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人気のある微分積分 >

integral of 11ln(x^2-x+2)

解

\int 11 ln (x^{2} - x + 2) d x

解

11 (x ln (x^{2} - x + 2) - \frac{1}{2} ln (x^{2} - x + 2) + 7 arctan (\frac{1}{7} (2 x - 1)) - 2 x + 1) + C

解答ステップ

\int 11 ln (x^{2} - x + 2) d x

定数を除く:

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

= 11 \cdot \int ln (x^{2} - x + 2) d x

平方完成する

x^{2} - x + 2 : (x - \frac{1}{2})^{2} + \frac{7}{4}

= 11 \cdot \int ln ((x - \frac{1}{2})^{2} + \frac{7}{4}) d x

u置換積分法を適用する

= 11 \cdot \int ln (u^{2} + \frac{7}{4}) d u

部分積分を適用する

= 11 (u ln (u^{2} + \frac{7}{4}) - \int \frac{8 u ^{2}}{4 u ^{2} + 7} d u)

\int \frac{8 u ^{2}}{4 u ^{2} + 7} d u = - 7 arctan (\frac{2}{7} u) + 2 u

= 11 (u ln (u^{2} + \frac{7}{4}) - (- 7 arctan (\frac{2}{7} u) + 2 u))

代用を戻す

u = x - \frac{1}{2}

= 11 ((x - \frac{1}{2}) ln ((x - \frac{1}{2})^{2} + \frac{7}{4}) - (- 7 arctan (\frac{2}{7} (x - \frac{1}{2})) + 2 (x - \frac{1}{2})))

簡素化

11 ((x - \frac{1}{2}) ln ((x - \frac{1}{2})^{2} + \frac{7}{4}) - (- 7 arctan (\frac{2}{7} (x - \frac{1}{2})) + 2 (x - \frac{1}{2}))) : 11 (x ln (x^{2} - x + 2) - \frac{1}{2} ln (x^{2} - x + 2) + 7 arctan (\frac{1}{7} (2 x - 1)) - 2 x + 1)

= 11 (x ln (x^{2} - x + 2) - \frac{1}{2} ln (x^{2} - x + 2) + 7 arctan (\frac{1}{7} (2 x - 1)) - 2 x + 1)

定数を解答に追加する

= 11 (x ln (x^{2} - x + 2) - \frac{1}{2} ln (x^{2} - x + 2) + 7 arctan (\frac{1}{7} (2 x - 1)) - 2 x + 1) + C

グラフ

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