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受欢迎的微积分 >

integral of (11x+17)/(2x^2+7x-4)

解答

\int \frac{11 x + 17}{2 x ^{2} + 7 x - 4} d x

解答

22 (\frac{1}{8} ln 1936 x^{2} + 6776 x - 3872 + \frac{1}{88} (ln \frac{4 x + 7}{9} + 1 - ln \frac{4 x + 7}{9} - 1)) + C

求解步骤

\int \frac{11 x + 17}{2 x ^{2} + 7 x - 4} d x

使用换元积分法

= \int \frac{11 u}{2 u ^{2} + 9 u - 1215} d u

提出常数:

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

= 11 \cdot \int \frac{u}{2 u ^{2} + 9 u - 1215} d u

对

2 u^{2} + 9 u - 1215

配方:

2 (u + \frac{9}{4})^{2} - \frac{9801}{8}

= 11 \cdot \int \frac{u}{2 ( u + \frac{9}{4} ) ^{2} - \frac{9801}{8}} d u

使用换元积分法

= 11 \cdot \int \frac{2 ( 4 v - 9 )}{16 v ^{2} - 9801} d v

提出常数:

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

= 11 \cdot 2 \cdot \int \frac{4 v - 9}{16 v ^{2} - 9801} d v

乘开

\frac{4 v - 9}{16 v ^{2} - 9801} : \frac{4 v}{16 v ^{2} - 9801} - \frac{9}{16 v ^{2} - 9801}

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 11 \cdot 2 (\int \frac{4 v}{16 v ^{2} - 9801} d v - \int \frac{9}{16 v ^{2} - 9801} d v)

\int \frac{4 v}{16 v ^{2} - 9801} d v = \frac{1}{8} ln 16 v^{2} - 9801

\int \frac{9}{16 v ^{2} - 9801} d v = - \frac{1}{88} (ln \frac{4}{99} v + 1 - ln \frac{4}{99} v - 1)

= 11 \cdot 2 (\frac{1}{8} ln 16 v^{2} - 9801 - (- \frac{1}{88} (ln \frac{4}{99} v + 1 - ln \frac{4}{99} v - 1)))

代回

= 11 \cdot 2 (\frac{1}{8} ln 16 (11 x + 17 + \frac{9}{4})^{2} - 9801 - (- \frac{1}{88} (ln \frac{4}{99} (11 x + 17 + \frac{9}{4}) + 1 - ln \frac{4}{99} (11 x + 17 + \frac{9}{4}) - 1)))

化简

11 \cdot 2 (\frac{1}{8} ln 16 (11 x + 17 + \frac{9}{4})^{2} - 9801 - (- \frac{1}{88} (ln \frac{4}{99} (11 x + 17 + \frac{9}{4}) + 1 - ln \frac{4}{99} (11 x + 17 + \frac{9}{4}) - 1))) : 22 (\frac{1}{8} ln 1936 x^{2} + 6776 x - 3872 + \frac{1}{88} (ln \frac{4 x + 7}{9} + 1 - ln \frac{4 x + 7}{9} - 1))

= 22 (\frac{1}{8} ln 1936 x^{2} + 6776 x - 3872 + \frac{1}{88} (ln \frac{4 x + 7}{9} + 1 - ln \frac{4 x + 7}{9} - 1))

解答补常数

= 22 (\frac{1}{8} ln 1936 x^{2} + 6776 x - 3872 + \frac{1}{88} (ln \frac{4 x + 7}{9} + 1 - ln \frac{4 x + 7}{9} - 1)) + C

作图

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