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受欢迎的微积分 >

integral of 13ln(x^2-x+6)

解答

\int 13 ln (x^{2} - x + 6) d x

解答

13 (x ln (x^{2} - x + 6) - \frac{1}{2} ln (x^{2} - x + 6) + 23 arctan (\frac{1}{23} (2 x - 1)) - 2 x + 1) + C

求解步骤

\int 13 ln (x^{2} - x + 6) d x

提出常数:

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

= 13 \cdot \int ln (x^{2} - x + 6) d x

对

x^{2} - x + 6

配方:

(x - \frac{1}{2})^{2} + \frac{23}{4}

= 13 \cdot \int ln ((x - \frac{1}{2})^{2} + \frac{23}{4}) d x

使用换元积分法

= 13 \cdot \int ln (u^{2} + \frac{23}{4}) d u

使用分布积分法

= 13 (u ln (u^{2} + \frac{23}{4}) - \int \frac{8 u ^{2}}{4 u ^{2} + 23} d u)

\int \frac{8 u ^{2}}{4 u ^{2} + 23} d u = - 23 arctan (\frac{2}{23} u) + 2 u

= 13 (u ln (u^{2} + \frac{23}{4}) - (- 23 arctan (\frac{2}{23} u) + 2 u))

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

代回

= 13 ((x - \frac{1}{2}) ln ((x - \frac{1}{2})^{2} + \frac{23}{4}) - (- 23 arctan (\frac{2}{23} (x - \frac{1}{2})) + 2 (x - \frac{1}{2})))

化简

13 ((x - \frac{1}{2}) ln ((x - \frac{1}{2})^{2} + \frac{23}{4}) - (- 23 arctan (\frac{2}{23} (x - \frac{1}{2})) + 2 (x - \frac{1}{2}))) : 13 (x ln (x^{2} - x + 6) - \frac{1}{2} ln (x^{2} - x + 6) + 23 arctan (\frac{1}{23} (2 x - 1)) - 2 x + 1)

= 13 (x ln (x^{2} - x + 6) - \frac{1}{2} ln (x^{2} - x + 6) + 23 arctan (\frac{1}{23} (2 x - 1)) - 2 x + 1)

解答补常数

= 13 (x ln (x^{2} - x + 6) - \frac{1}{2} ln (x^{2} - x + 6) + 23 arctan (\frac{1}{23} (2 x - 1)) - 2 x + 1) + C

作图

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