integral of-(4x+10)/(f^2(x)*x^3)

解答

\int - \frac{4 x + 10}{f ^{2} ( x ) \cdot x ^{3}}

- \frac{1}{f ^{2}} (- \frac{10}{3 x ^{3}} - \frac{2}{x ^{2}}) + C

求解步骤

\int - \frac{4 x + 10}{f ^{2} x x ^{3}} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{f ^{2}} \cdot \int \frac{10 + 4 x}{x x ^{3}} d x

乘开

\frac{10 + 4 x}{x x ^{3}} : \frac{10}{x ^{4}} + \frac{4}{x ^{3}}

= - \frac{1}{f ^{2}} \cdot \int \frac{10}{x ^{4}} + \frac{4}{x ^{3}} 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

= - \frac{1}{f ^{2}} (\int \frac{10}{x ^{4}} d x + \int \frac{4}{x ^{3}} d x)

\int \frac{10}{x ^{4}} d x = - \frac{10}{3 x ^{3}}

\int \frac{4}{x ^{3}} d x = - \frac{2}{x ^{2}}

= - \frac{1}{f ^{2}} (- \frac{10}{3 x ^{3}} - \frac{2}{x ^{2}})

解答补常数

= - \frac{1}{f ^{2}} (- \frac{10}{3 x ^{3}} - \frac{2}{x ^{2}}) + C

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