integral of 1/(x^2(x^2+5))

解答

\int \frac{1}{x ^{2} ( x ^{2} + 5 )} d x

- \frac{1}{5 x} - \frac{1}{5 5} arctan (\frac{x}{5}) + C

求解步骤

\int \frac{1}{x ^{2} ( x ^{2} + 5 )} d x

将

\frac{1}{x ^{2} ( x ^{2} + 5 )}

用部份分式展开:

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

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

\int \frac{1}{5 x ^{2}} d x = - \frac{1}{5 x}

\int \frac{1}{5 ( x ^{2} + 5 )} d x = \frac{1}{5 5} arctan (\frac{x}{5})

= - \frac{1}{5 x} - \frac{1}{5 5} arctan (\frac{x}{5})

解答补常数

= - \frac{1}{5 x} - \frac{1}{5 5} arctan (\frac{x}{5}) + C