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受欢迎的微积分 >

integral of (x^3)/(sqrt(x^2+2x+5))

解答

\int \frac{x ^{3}}{x ^{2} + 2 x + 5} d x

解答

\frac{1}{3} (x^{2} + 2 x + 5)^{\frac{3}{2}} - x^{2} + 2 x + 5 + 5 ln (\frac{1}{2} x + x^{2} + 2 x + 5 + 1) - \frac{3}{2} (x + 1) x^{2} + 2 x + 5 + C

求解步骤

\int \frac{x ^{3}}{x ^{2} + 2 x + 5} d x

对

x^{2} + 2 x + 5

配方:

(x + 1)^{2} + 4

= \int \frac{x ^{3}}{( x + 1 ) ^{2} + 4} d x

使用换元积分法

= \int \frac{( u - 1 ) ^{3}}{u ^{2} + 4} d u

乘开

\frac{( u - 1 ) ^{3}}{u ^{2} + 4} : \frac{u ^{3}}{u ^{2} + 4} - \frac{3 u ^{2}}{u ^{2} + 4} + \frac{3 u}{u ^{2} + 4} - \frac{1}{u ^{2} + 4}

= \int \frac{u ^{3}}{u ^{2} + 4} - \frac{3 u ^{2}}{u ^{2} + 4} + \frac{3 u}{u ^{2} + 4} - \frac{1}{u ^{2} + 4} 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 \frac{u ^{3}}{u ^{2} + 4} d u - \int \frac{3 u ^{2}}{u ^{2} + 4} d u + \int \frac{3 u}{u ^{2} + 4} d u - \int \frac{1}{u ^{2} + 4} d u

\int \frac{u ^{3}}{u ^{2} + 4} d u = \frac{1}{3} (u^{2} + 4)^{\frac{3}{2}} - 4 u^{2} + 4

\int \frac{3 u ^{2}}{u ^{2} + 4} d u = 12 (- \frac{1}{2} ln (\frac{u + 4 + u ^{2}}{2}) + \frac{1}{8} u 4 + u^{2})

\int \frac{3 u}{u ^{2} + 4} d u = 3 u^{2} + 4

\int \frac{1}{u ^{2} + 4} d u = ln (\frac{1}{2} u + 4 + u^{2})

= \frac{1}{3} (u^{2} + 4)^{\frac{3}{2}} - 4 u^{2} + 4 - 12 (- \frac{1}{2} ln (\frac{u + 4 + u ^{2}}{2}) + \frac{1}{8} u 4 + u^{2}) + 3 u^{2} + 4 - ln (\frac{1}{2} u + 4 + u^{2})

u = x + 1

代回

= \frac{1}{3} ((x + 1)^{2} + 4)^{\frac{3}{2}} - 4 (x + 1)^{2} + 4 - 12 - \frac{1}{2} ln \frac{x + 1 + 4 + ( x + 1 ) ^{2}}{2} + \frac{1}{8} (x + 1) 4 + (x + 1)^{2} + 3 (x + 1)^{2} + 4 - ln (\frac{1}{2} x + 1 + 4 + (x + 1)^{2})

化简

\frac{1}{3} ((x + 1)^{2} + 4)^{\frac{3}{2}} - 4 (x + 1)^{2} + 4 - 12 - \frac{1}{2} ln \frac{x + 1 + 4 + ( x + 1 ) ^{2}}{2} + \frac{1}{8} (x + 1) 4 + (x + 1)^{2} + 3 (x + 1)^{2} + 4 - ln (\frac{1}{2} x + 1 + 4 + (x + 1)^{2}) : \frac{1}{3} (x^{2} + 2 x + 5)^{\frac{3}{2}} - x^{2} + 2 x + 5 + 5 ln (\frac{1}{2} x + x^{2} + 2 x + 5 + 1) - \frac{3}{2} (x + 1) x^{2} + 2 x + 5

= \frac{1}{3} (x^{2} + 2 x + 5)^{\frac{3}{2}} - x^{2} + 2 x + 5 + 5 ln (\frac{1}{2} x + x^{2} + 2 x + 5 + 1) - \frac{3}{2} (x + 1) x^{2} + 2 x + 5

解答补常数

= \frac{1}{3} (x^{2} + 2 x + 5)^{\frac{3}{2}} - x^{2} + 2 x + 5 + 5 ln (\frac{1}{2} x + x^{2} + 2 x + 5 + 1) - \frac{3}{2} (x + 1) x^{2} + 2 x + 5 + C

作图

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