integral of (x^2)/(\sqrt[3]{x+2)}

解答

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

\frac{3}{8} (x + 2)^{\frac{8}{3}} - \frac{12}{5} (x + 2)^{\frac{5}{3}} + 6 (x + 2)^{\frac{2}{3}} + C

求解步骤

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

使用换元积分法

= \int \frac{( u - 2 ) ^{2}}{3 u} d u

乘开

\frac{( u - 2 ) ^{2}}{3 u} : u^{\frac{5}{3}} - 4 u^{\frac{2}{3}} + \frac{4}{3 u}

= \int u^{\frac{5}{3}} - 4 u^{\frac{2}{3}} + \frac{4}{3 u} d u

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 u^{\frac{5}{3}} d u - \int 4 u^{\frac{2}{3}} d u + \int \frac{4}{3 u} d u

\int u^{\frac{5}{3}} d u = \frac{3}{8} u^{\frac{8}{3}}

\int 4 u^{\frac{2}{3}} d u = \frac{12}{5} u^{\frac{5}{3}}

\int \frac{4}{3 u} d u = 6 u^{\frac{2}{3}}

= \frac{3}{8} u^{\frac{8}{3}} - \frac{12}{5} u^{\frac{5}{3}} + 6 u^{\frac{2}{3}}

u = x + 2

代回

= \frac{3}{8} (x + 2)^{\frac{8}{3}} - \frac{12}{5} (x + 2)^{\frac{5}{3}} + 6 (x + 2)^{\frac{2}{3}}

解答补常数

= \frac{3}{8} (x + 2)^{\frac{8}{3}} - \frac{12}{5} (x + 2)^{\frac{5}{3}} + 6 (x + 2)^{\frac{2}{3}} + C

\int - \frac{2}{81 - 49 t ^{2}} d t

\frac{d}{d x} (\frac{x ^{6}}{x ^{2}})

\int \frac{x ^{2} - 9}{x} d x

d er i v a t i v e y = \frac{1}{x ^{2} + 1}

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