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受欢迎的微积分 >

integral of ((x^2))/(sqrt(13-4x+x^2))

解答

\int \frac{( x ^{2} )}{13 - 4 x + x ^{2}} d x

解答

9 (- \frac{1}{2} ln (\frac{x - 2 + x ^{2} - 4 x + 13}{3}) + \frac{1}{18} (x - 2) x^{2} - 4 x + 13) + 4 x^{2} - 4 x + 13 + 4 ln (\frac{1}{3} x - 2 + x^{2} - 4 x + 13) + C

求解步骤

\int \frac{x ^{2}}{13 - 4 x + x ^{2}} d x

对

13 - 4 x + x^{2}

配方:

(x - 2)^{2} + 9

= \int \frac{x ^{2}}{( x - 2 ) ^{2} + 9} d x

使用换元积分法

= \int \frac{( u + 2 ) ^{2}}{u ^{2} + 9} d u

乘开

\frac{( u + 2 ) ^{2}}{u ^{2} + 9} : \frac{u ^{2}}{u ^{2} + 9} + \frac{4 u}{u ^{2} + 9} + \frac{4}{u ^{2} + 9}

= \int \frac{u ^{2}}{u ^{2} + 9} + \frac{4 u}{u ^{2} + 9} + \frac{4}{u ^{2} + 9} 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 ^{2}}{u ^{2} + 9} d u + \int \frac{4 u}{u ^{2} + 9} d u + \int \frac{4}{u ^{2} + 9} d u

\int \frac{u ^{2}}{u ^{2} + 9} d u = 9 (- \frac{1}{2} ln (\frac{u + 9 + u ^{2}}{3}) + \frac{1}{18} u 9 + u^{2})

\int \frac{4 u}{u ^{2} + 9} d u = 4 u^{2} + 9

\int \frac{4}{u ^{2} + 9} d u = 4 ln (\frac{1}{3} u + 9 + u^{2})

= 9 (- \frac{1}{2} ln (\frac{u + 9 + u ^{2}}{3}) + \frac{1}{18} u 9 + u^{2}) + 4 u^{2} + 9 + 4 ln (\frac{1}{3} u + 9 + u^{2})

u = x - 2

代回

= 9 - \frac{1}{2} ln \frac{x - 2 + 9 + ( x - 2 ) ^{2}}{3} + \frac{1}{18} (x - 2) 9 + (x - 2)^{2} + 4 (x - 2)^{2} + 9 + 4 ln (\frac{1}{3} x - 2 + 9 + (x - 2)^{2})

化简

9 - \frac{1}{2} ln \frac{x - 2 + 9 + ( x - 2 ) ^{2}}{3} + \frac{1}{18} (x - 2) 9 + (x - 2)^{2} + 4 (x - 2)^{2} + 9 + 4 ln (\frac{1}{3} x - 2 + 9 + (x - 2)^{2}) : 9 (- \frac{1}{2} ln (\frac{x - 2 + x ^{2} - 4 x + 13}{3}) + \frac{1}{18} (x - 2) x^{2} - 4 x + 13) + 4 x^{2} - 4 x + 13 + 4 ln (\frac{1}{3} x - 2 + x^{2} - 4 x + 13)

= 9 (- \frac{1}{2} ln (\frac{x - 2 + x ^{2} - 4 x + 13}{3}) + \frac{1}{18} (x - 2) x^{2} - 4 x + 13) + 4 x^{2} - 4 x + 13 + 4 ln (\frac{1}{3} x - 2 + x^{2} - 4 x + 13)

解答补常数

= 9 (- \frac{1}{2} ln (\frac{x - 2 + x ^{2} - 4 x + 13}{3}) + \frac{1}{18} (x - 2) x^{2} - 4 x + 13) + 4 x^{2} - 4 x + 13 + 4 ln (\frac{1}{3} x - 2 + x^{2} - 4 x + 13) + C

作图

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