integral of (7x^2+x+54)/(x^3+9x)

解答

\int \frac{7 x ^{2} + x + 54}{x ^{3} + 9 x} d x

6 ln ∣ x ∣ + \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3}) + C

求解步骤

\int \frac{7 x ^{2} + x + 54}{x ^{3} + 9 x} d x

将

\frac{7 x ^{2} + x + 54}{x ^{3} + 9 x}

用部份分式展开:

\frac{6}{x} + \frac{x + 1}{x ^{2} + 9}

= \int \frac{6}{x} + \frac{x + 1}{x ^{2} + 9} d 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{6}{x} d x + \int \frac{x + 1}{x ^{2} + 9} d x

\int \frac{6}{x} d x = 6 ln ∣ x ∣

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

= 6 ln ∣ x ∣ + \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3})

解答补常数

= 6 ln ∣ x ∣ + \frac{1}{2} ln x^{2} + 9 + \frac{1}{3} arctan (\frac{x}{3}) + C

d er i v a t i v e y = x + 2

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