integral of (x^{1/3}-3)/(x^{2/3)}

解答

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

\frac{3}{2} x^{\frac{2}{3}} - 9 x^{\frac{1}{3}} + C

求解步骤

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

乘开

\frac{x ^{\frac{1}{3}} - 3}{x ^{\frac{2}{3}}} : \frac{1}{x ^{\frac{1}{3}}} - \frac{3}{x ^{\frac{2}{3}}}

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

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

\int \frac{3}{x ^{\frac{2}{3}}} d x = 9 x^{\frac{1}{3}}

= \frac{3}{2} x^{\frac{2}{3}} - 9 x^{\frac{1}{3}}

解答补常数

= \frac{3}{2} x^{\frac{2}{3}} - 9 x^{\frac{1}{3}} + C

\int_{0}^{2} 1 + 4 x^{2} d x

q^{^{'}} + \frac{q}{20 + t} = 1, q (0) = 5

d er i v a t i v e x^{8} (x - 3)^{7}

n = 0 \sum \infty \frac{5}{4 ^{n + 1}}

x \to o lim (\frac{sin ( 4 x )}{x})