integral of (x+2)/(x^2-25)

解答

\int \frac{x + 2}{x ^{2} - 25} d x

\frac{1}{2} ln x^{2} - 25 - \frac{1}{5} (ln \frac{x}{5} + 1 - ln \frac{x}{5} - 1) + C

求解步骤

\int \frac{x + 2}{x ^{2} - 25} d x

乘开

\frac{x + 2}{x ^{2} - 25} : \frac{x}{x ^{2} - 25} + \frac{2}{x ^{2} - 25}

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}{x ^{2} - 25} d x + \int \frac{2}{x ^{2} - 25} d x

\int \frac{x}{x ^{2} - 25} d x = \frac{1}{2} ln x^{2} - 25

\int \frac{2}{x ^{2} - 25} d x = - \frac{1}{5} (ln \frac{x}{5} + 1 - ln \frac{x}{5} - 1)

= \frac{1}{2} ln x^{2} - 25 - \frac{1}{5} (ln \frac{x}{5} + 1 - ln \frac{x}{5} - 1)

解答补常数

= \frac{1}{2} ln x^{2} - 25 - \frac{1}{5} (ln \frac{x}{5} + 1 - ln \frac{x}{5} - 1) + C

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