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integral of (x^3+24)/(x^2+7x+12)

解答

\int \frac{x ^{3} + 24}{x ^{2} + 7 x + 12} d x

解答

\frac{1}{8} (2 x + 7)^{2} - \frac{21}{2} (x + \frac{7}{2}) + 4 (\frac{37}{8} ln 4 x^{2} + 28 x + 48 + \frac{43}{8} ln ∣ 2 x + 8 ∣ - \frac{43}{8} ln ∣ 2 x + 6 ∣) + C

求解步骤

\int \frac{x ^{3} + 24}{x ^{2} + 7 x + 12} d x

长除

\frac{x ^{3} + 24}{x ^{2} + 7 x + 12} : x - 7 + \frac{37 x + 108}{x ^{2} + 7 x + 12}

= \int x - 7 + \frac{37 x + 108}{x ^{2} + 7 x + 12} d x

对

x^{2} + 7 x + 12

配方:

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

= \int x - 7 + \frac{37 x + 108}{( x + \frac{7}{2} ) ^{2} - \frac{1}{4}} d x

使用换元积分法

= \int u - \frac{7}{2} - 7 + \frac{4 ( 37 ( u - \frac{7}{2} ) + 108 )}{4 u ^{2} - 1} d u

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

= \int u d u - \int \frac{7}{2} d u - \int 7 d u + \int \frac{4 ( 37 ( u - \frac{7}{2} ) + 108 )}{4 u ^{2} - 1} d u

\int u d u = \frac{u ^{2}}{2}

\int \frac{7}{2} d u = \frac{7}{2} u

\int 7 d u = 7 u

\int \frac{4 ( 37 ( u - \frac{7}{2} ) + 108 )}{4 u ^{2} - 1} d u = 4 (\frac{37}{8} ln 4 u^{2} - 1 + \frac{43}{8} ln ∣ 2 u + 1 ∣ - \frac{43}{8} ln ∣ 2 u - 1 ∣)

= \frac{u ^{2}}{2} - \frac{7}{2} u - 7 u + 4 (\frac{37}{8} ln 4 u^{2} - 1 + \frac{43}{8} ln ∣ 2 u + 1 ∣ - \frac{43}{8} ln ∣ 2 u - 1 ∣)

u = x + \frac{7}{2}

代回

= \frac{( x + \frac{7}{2} ) ^{2}}{2} - \frac{7}{2} (x + \frac{7}{2}) - 7 (x + \frac{7}{2}) + 4 (\frac{37}{8} ln 4 (x + \frac{7}{2})^{2} - 1 + \frac{43}{8} ln 2 (x + \frac{7}{2}) + 1 - \frac{43}{8} ln 2 (x + \frac{7}{2}) - 1)

化简

\frac{( x + \frac{7}{2} ) ^{2}}{2} - \frac{7}{2} (x + \frac{7}{2}) - 7 (x + \frac{7}{2}) + 4 (\frac{37}{8} ln 4 (x + \frac{7}{2})^{2} - 1 + \frac{43}{8} ln 2 (x + \frac{7}{2}) + 1 - \frac{43}{8} ln 2 (x + \frac{7}{2}) - 1) : \frac{1}{8} (2 x + 7)^{2} - \frac{21}{2} (x + \frac{7}{2}) + 4 (\frac{37}{8} ln 4 x^{2} + 28 x + 48 + \frac{43}{8} ln ∣ 2 x + 8 ∣ - \frac{43}{8} ln ∣ 2 x + 6 ∣)

= \frac{1}{8} (2 x + 7)^{2} - \frac{21}{2} (x + \frac{7}{2}) + 4 (\frac{37}{8} ln 4 x^{2} + 28 x + 48 + \frac{43}{8} ln ∣ 2 x + 8 ∣ - \frac{43}{8} ln ∣ 2 x + 6 ∣)

解答补常数

= \frac{1}{8} (2 x + 7)^{2} - \frac{21}{2} (x + \frac{7}{2}) + 4 (\frac{37}{8} ln 4 x^{2} + 28 x + 48 + \frac{43}{8} ln ∣ 2 x + 8 ∣ - \frac{43}{8} ln ∣ 2 x + 6 ∣) + C

作图

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