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受欢迎的微积分 >

integral of (8x^2-7x+8)/(x^3+4x)

解答

\int \frac{8 x ^{2} - 7 x + 8}{x ^{3} + 4 x} d x

解答

- \frac{7}{2} arctan (\frac{x}{2}) + 3 ln x^{2} + 4 + 2 ln ∣ x ∣ + C

求解步骤

\int \frac{8 x ^{2} - 7 x + 8}{x ^{3} + 4 x} d x

乘开

\frac{8 x ^{2} - 7 x + 8}{x ^{3} + 4 x} : \frac{8 x ^{2}}{x ^{3} + 4 x} - \frac{7 x}{x ^{3} + 4 x} + \frac{8}{x ^{3} + 4 x}

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{8 x ^{2}}{x ^{3} + 4 x} d x - \int \frac{7 x}{x ^{3} + 4 x} d x + \int \frac{8}{x ^{3} + 4 x} d x

\int \frac{8 x ^{2}}{x ^{3} + 4 x} d x = 4 ln x^{2} + 4

\int \frac{7 x}{x ^{3} + 4 x} d x = \frac{7}{2} arctan (\frac{x}{2})

\int \frac{8}{x ^{3} + 4 x} d x = 2 ln ∣ x ∣ - ln x^{2} + 4

= 4 ln x^{2} + 4 - \frac{7}{2} arctan (\frac{x}{2}) + 2 ln ∣ x ∣ - ln x^{2} + 4

化简

= - \frac{7}{2} arctan (\frac{x}{2}) + 3 ln x^{2} + 4 + 2 ln ∣ x ∣

解答补常数

= - \frac{7}{2} arctan (\frac{x}{2}) + 3 ln x^{2} + 4 + 2 ln ∣ x ∣ + C

作图

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