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

解答

\int \frac{( x + 3 x ) ^{3}}{x} d x

\frac{2}{3} x^{\frac{3}{2}} + \frac{9}{4} x^{\frac{4}{3}} + \frac{18}{7} x^{\frac{7}{6}} + x + C

求解步骤

\int \frac{( x + 3 x ) ^{3}}{x} d x

乘开

\frac{( x + 3 x ) ^{3}}{x} : x^{\frac{1}{2}} + 3 x^{\frac{1}{3}} + 3 x^{\frac{1}{6}} + 1

= \int x^{\frac{1}{2}} + 3 x^{\frac{1}{3}} + 3 x^{\frac{1}{6}} + 1 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 x^{\frac{1}{2}} d x + \int 3 x^{\frac{1}{3}} d x + \int 3 x^{\frac{1}{6}} d x + \int 1 d x

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

\int 3 x^{\frac{1}{3}} d x = \frac{9}{4} x^{\frac{4}{3}}

\int 3 x^{\frac{1}{6}} d x = \frac{18}{7} x^{\frac{7}{6}}

\int 1 d x = x

= \frac{2}{3} x^{\frac{3}{2}} + \frac{9}{4} x^{\frac{4}{3}} + \frac{18}{7} x^{\frac{7}{6}} + x

解答补常数

= \frac{2}{3} x^{\frac{3}{2}} + \frac{9}{4} x^{\frac{4}{3}} + \frac{18}{7} x^{\frac{7}{6}} + x + C

\int \frac{e ^{t}}{1 - e ^{2 t}} d t

n = 0 \sum \infty (- 1)^{n} (n + 1)

\int x e^{(- x)} d x

d er i v a t i v e f (x) = e^{l n (x)} - e^{5 l n (x^{2})}

d er i v a t i v e f (x) = tan (x) cot (x)