integral of ((5x^2+8x+5))/((x-5)(x^2+9))

解答

\int \frac{( 5 x ^{2} + 8 x + 5 )}{( x - 5 ) ( x ^{2} + 9 )} d x

5 ln ∣ x - 5 ∣ + \frac{10}{17} ln x^{2} + 9 + \frac{8}{3} arctan (\frac{x}{3}) - \frac{10}{17} ln x^{2} + 9 + C

求解步骤

\int \frac{5 x ^{2} + 8 x + 5}{( x - 5 ) ( x ^{2} + 9 )} d x

乘开

\frac{5 x ^{2} + 8 x + 5}{( x - 5 ) ( x ^{2} + 9 )} : \frac{5 x ^{2}}{( x - 5 ) ( x ^{2} + 9 )} + \frac{8 x}{( x - 5 ) ( x ^{2} + 9 )} + \frac{5}{( x - 5 ) ( x ^{2} + 9 )}

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{5 x ^{2}}{( x - 5 ) ( x ^{2} + 9 )} d x + \int \frac{8 x}{( x - 5 ) ( x ^{2} + 9 )} d x + \int \frac{5}{( x - 5 ) ( x ^{2} + 9 )} d x

\int \frac{5 x ^{2}}{( x - 5 ) ( x ^{2} + 9 )} d x = 5 (\frac{25}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (\frac{9}{2} ln x^{2} + 9 + 15 arctan (\frac{x}{3})))

\int \frac{8 x}{( x - 5 ) ( x ^{2} + 9 )} d x = 8 (\frac{5}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (- \frac{5}{2} ln x^{2} + 9 + 3 arctan (\frac{x}{3})))

\int \frac{5}{( x - 5 ) ( x ^{2} + 9 )} d x = 5 (\frac{1}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (- \frac{1}{2} ln x^{2} + 9 - \frac{5}{3} arctan (\frac{x}{3})))

= 5 (\frac{25}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (\frac{9}{2} ln x^{2} + 9 + 15 arctan (\frac{x}{3}))) + 8 (\frac{5}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (- \frac{5}{2} ln x^{2} + 9 + 3 arctan (\frac{x}{3}))) + 5 (\frac{1}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (- \frac{1}{2} ln x^{2} + 9 - \frac{5}{3} arctan (\frac{x}{3})))

化简

5 (\frac{25}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (\frac{9}{2} ln x^{2} + 9 + 15 arctan (\frac{x}{3}))) + 8 (\frac{5}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (- \frac{5}{2} ln x^{2} + 9 + 3 arctan (\frac{x}{3}))) + 5 (\frac{1}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (- \frac{1}{2} ln x^{2} + 9 - \frac{5}{3} arctan (\frac{x}{3}))) : 5 ln ∣ x - 5 ∣ + \frac{10}{17} ln x^{2} + 9 + \frac{8}{3} arctan (\frac{x}{3}) - \frac{10}{17} ln x^{2} + 9

= 5 ln ∣ x - 5 ∣ + \frac{10}{17} ln x^{2} + 9 + \frac{8}{3} arctan (\frac{x}{3}) - \frac{10}{17} ln x^{2} + 9

解答补常数

= 5 ln ∣ x - 5 ∣ + \frac{10}{17} ln x^{2} + 9 + \frac{8}{3} arctan (\frac{x}{3}) - \frac{10}{17} ln x^{2} + 9 + C

\int 2 sin (x^{3}) d x

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\frac{d}{d x} (\frac{3 x}{2 sin ( x ) + cos ( x )})

y^{^{''}} + y^{^{'}} - 6 y = 0

n = 1 \sum \infty \frac{5 ^{n}}{6 ^{n}}