integral of ((3x^3-4x-2))/(x^3)

解答

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

3 x + \frac{4}{x} + \frac{1}{x ^{2}} + C

求解步骤

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

乘开

\frac{3 x ^{3} - 4 x - 2}{x ^{3}} : 3 - \frac{4}{x ^{2}} - \frac{2}{x ^{3}}

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

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

\int 3 d x = 3 x

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

\int \frac{2}{x ^{3}} d x = - \frac{1}{x ^{2}}

= 3 x - (- \frac{4}{x}) - (- \frac{1}{x ^{2}})

化简

= 3 x + \frac{4}{x} + \frac{1}{x ^{2}}

解答补常数

= 3 x + \frac{4}{x} + \frac{1}{x ^{2}} + C

\int x^{2} + sec^{2} (x) d x

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

\int e^{(2 - s) t} d t

\int tan (\frac{x}{2}) sec^{3} (\frac{x}{2}) d x

\int \frac{30}{x} d x