integral of (3sqrt(x)+7)/(\sqrt[3]{x)}

解答

\int \frac{3 x + 7}{3 x} d x

\frac{18}{7} x^{\frac{7}{6}} + \frac{21}{2} x^{\frac{2}{3}} + C

求解步骤

\int \frac{3 x + 7}{3 x} d x

乘开

\frac{3 x + 7}{3 x} : 3 6 x + \frac{7}{3 x}

= \int 3 6 x + \frac{7}{3 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

= \int 3 6 x d x + \int \frac{7}{3 x} d x

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

\int \frac{7}{3 x} d x = \frac{21}{2} x^{\frac{2}{3}}

= \frac{18}{7} x^{\frac{7}{6}} + \frac{21}{2} x^{\frac{2}{3}}

解答补常数

= \frac{18}{7} x^{\frac{7}{6}} + \frac{21}{2} x^{\frac{2}{3}} + C

\int x^{3} (3 - x^{4})^{10} d x

\int x (6 x + 7)^{8} d x

\int x \cdot x^{10} d x

\frac{d y}{d x} y = x^{8 c o s (x)}

x \to 0 lim (\frac{sin ( 5 x )}{sin ( \frac{1}{3} x )})