integral of (x^4-\sqrt[3]{x})/(6sqrt(x))

解答

\int \frac{x ^{4} - 3 x}{6 x} d x

\frac{1}{135} (5 x^{\frac{9}{2}} - 27 x^{\frac{5}{6}}) + C

求解步骤

\int \frac{x ^{4} - 3 x}{6 x} d x

提出常数:

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

= \frac{1}{6} \cdot \int \frac{x ^{4} - 3 x}{x} d x

乘开

\frac{x ^{4} - 3 x}{x} : x^{\frac{7}{2}} - \frac{1}{6 x}

= \frac{1}{6} \cdot \int x^{\frac{7}{2}} - \frac{1}{6 x} 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}{6} (\int x^{\frac{7}{2}} d x - \int \frac{1}{6 x} d x)

\int x^{\frac{7}{2}} d x = \frac{2}{9} x^{\frac{9}{2}}

\int \frac{1}{6 x} d x = \frac{6}{5} x^{\frac{5}{6}}

= \frac{1}{6} (\frac{2}{9} x^{\frac{9}{2}} - \frac{6}{5} x^{\frac{5}{6}})

化简

\frac{1}{6} (\frac{2}{9} x^{\frac{9}{2}} - \frac{6}{5} x^{\frac{5}{6}}) : \frac{1}{135} (5 x^{\frac{9}{2}} - 27 x^{\frac{5}{6}})

= \frac{1}{135} (5 x^{\frac{9}{2}} - 27 x^{\frac{5}{6}})

解答补常数

= \frac{1}{135} (5 x^{\frac{9}{2}} - 27 x^{\frac{5}{6}}) + C

\int \frac{1}{x} ln (x) d x

\int - (t - 6)^{2} d t

\frac{d}{d y} (x^{0.5} y^{0.5})

d er i v a t i v e x^{\frac{7}{9}}

t an g e n t f (x) = x^{2} - 7, at x = 2