integral of (4x-1)/(\sqrt[3]{3x+1)}

解答

\int \frac{4 x - 1}{3 3 x + 1} d x

4 (\frac{1}{2} x (3 x + 1)^{\frac{2}{3}} - \frac{1}{10} (3 x + 1)^{\frac{5}{3}}) - \frac{1}{2} (3 x + 1)^{\frac{2}{3}} + C

求解步骤

\int \frac{4 x - 1}{3 3 x + 1} d x

乘开

\frac{4 x - 1}{3 3 x + 1} : \frac{4 x}{3 3 x + 1} - \frac{1}{3 3 x + 1}

= \int \frac{4 x}{3 3 x + 1} - \frac{1}{3 3 x + 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 \frac{4 x}{3 3 x + 1} d x - \int \frac{1}{3 3 x + 1} d x

\int \frac{4 x}{3 3 x + 1} d x = 4 (\frac{1}{2} x (3 x + 1)^{\frac{2}{3}} - \frac{1}{10} (3 x + 1)^{\frac{5}{3}})

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

= 4 (\frac{1}{2} x (3 x + 1)^{\frac{2}{3}} - \frac{1}{10} (3 x + 1)^{\frac{5}{3}}) - \frac{1}{2} (3 x + 1)^{\frac{2}{3}}

解答补常数

= 4 (\frac{1}{2} x (3 x + 1)^{\frac{2}{3}} - \frac{1}{10} (3 x + 1)^{\frac{5}{3}}) - \frac{1}{2} (3 x + 1)^{\frac{2}{3}} + C

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