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受欢迎的微积分 >

integral of 15ln(x^2-x+12)

解答

\int 15 ln (x^{2} - x + 12) d x

解答

15 (x ln (x^{2} - x + 12) - \frac{1}{2} ln (x^{2} - x + 12) + 47 arctan (\frac{1}{47} (2 x - 1)) - 2 x + 1) + C

求解步骤

\int 15 ln (x^{2} - x + 12) d x

提出常数:

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

= 15 \cdot \int ln (x^{2} - x + 12) d x

对

x^{2} - x + 12

配方:

(x - \frac{1}{2})^{2} + \frac{47}{4}

= 15 \cdot \int ln ((x - \frac{1}{2})^{2} + \frac{47}{4}) d x

使用换元积分法

= 15 \cdot \int ln (u^{2} + \frac{47}{4}) d u

使用分布积分法

= 15 (u ln (u^{2} + \frac{47}{4}) - \int \frac{8 u ^{2}}{4 u ^{2} + 47} d u)

\int \frac{8 u ^{2}}{4 u ^{2} + 47} d u = - 47 arctan (\frac{2}{47} u) + 2 u

= 15 (u ln (u^{2} + \frac{47}{4}) - (- 47 arctan (\frac{2}{47} u) + 2 u))

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

代回

= 15 ((x - \frac{1}{2}) ln ((x - \frac{1}{2})^{2} + \frac{47}{4}) - (- 47 arctan (\frac{2}{47} (x - \frac{1}{2})) + 2 (x - \frac{1}{2})))

化简

15 ((x - \frac{1}{2}) ln ((x - \frac{1}{2})^{2} + \frac{47}{4}) - (- 47 arctan (\frac{2}{47} (x - \frac{1}{2})) + 2 (x - \frac{1}{2}))) : 15 (x ln (x^{2} - x + 12) - \frac{1}{2} ln (x^{2} - x + 12) + 47 arctan (\frac{1}{47} (2 x - 1)) - 2 x + 1)

= 15 (x ln (x^{2} - x + 12) - \frac{1}{2} ln (x^{2} - x + 12) + 47 arctan (\frac{1}{47} (2 x - 1)) - 2 x + 1)

解答补常数

= 15 (x ln (x^{2} - x + 12) - \frac{1}{2} ln (x^{2} - x + 12) + 47 arctan (\frac{1}{47} (2 x - 1)) - 2 x + 1) + C

作图

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