integral of (x^2+4)/((x-5)(x^2+9))

解答

\int \frac{x ^{2} + 4}{( x - 5 ) ( x ^{2} + 9 )} d x

\frac{9}{68} ln x^{2} + 9 + \frac{29}{34} ln ∣ x - 5 ∣ + \frac{15}{34} arctan (\frac{x}{3}) + \frac{1}{51} (- 3 ln x^{2} + 9 - 10 arctan (\frac{x}{3})) + C

求解步骤

\int \frac{x ^{2} + 4}{( x - 5 ) ( x ^{2} + 9 )} d x

乘开

\frac{x ^{2} + 4}{( x - 5 ) ( x ^{2} + 9 )} : \frac{x ^{2}}{( x - 5 ) ( x ^{2} + 9 )} + \frac{4}{( 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{x ^{2}}{( x - 5 ) ( x ^{2} + 9 )} d x + \int \frac{4}{( x - 5 ) ( x ^{2} + 9 )} d x

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

\int \frac{4}{( x - 5 ) ( x ^{2} + 9 )} d x = 4 (\frac{1}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (- \frac{1}{2} ln x^{2} + 9 - \frac{5}{3} arctan (\frac{x}{3})))

= \frac{25}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (\frac{9}{2} ln x^{2} + 9 + 15 arctan (\frac{x}{3})) + 4 (\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

\frac{25}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (\frac{9}{2} ln x^{2} + 9 + 15 arctan (\frac{x}{3})) + 4 (\frac{1}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (- \frac{1}{2} ln x^{2} + 9 - \frac{5}{3} arctan (\frac{x}{3}))) : \frac{9}{68} ln x^{2} + 9 + \frac{29}{34} ln ∣ x - 5 ∣ + \frac{15}{34} arctan (\frac{x}{3}) + \frac{1}{51} (- 3 ln x^{2} + 9 - 10 arctan (\frac{x}{3}))

= \frac{9}{68} ln x^{2} + 9 + \frac{29}{34} ln ∣ x - 5 ∣ + \frac{15}{34} arctan (\frac{x}{3}) + \frac{1}{51} (- 3 ln x^{2} + 9 - 10 arctan (\frac{x}{3}))

解答补常数

= \frac{9}{68} ln x^{2} + 9 + \frac{29}{34} ln ∣ x - 5 ∣ + \frac{15}{34} arctan (\frac{x}{3}) + \frac{1}{51} (- 3 ln x^{2} + 9 - 10 arctan (\frac{x}{3})) + C

x \to 0 lim (ln (x) \cdot x)

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

\int (\frac{x}{3})^{n} d x

d er i v a t i v e (x^{4} - 3)^{3} (x^{2} + 6)^{5}

\frac{d}{d x} (4 x^{\frac{5}{4}} + 2 x^{\frac{3}{2}} + 5 x)