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

解答

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

x + \frac{3}{5} x^{\frac{5}{3}} - \frac{3}{4} x^{\frac{4}{3}} - \frac{3}{2} x^{\frac{2}{3}} - 3 3 x + \frac{3}{2} (3 x + 1)^{2} - 6 + C

求解步骤

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

乘开

\frac{1 + x}{1 + 3 x} : \frac{1}{1 + 3 x} + \frac{x}{1 + 3 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}{1 + 3 x} d x + \int \frac{x}{1 + 3 x} d x

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

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

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

化简

3 (\frac{( 1 + 3 x ) ^{2}}{2} - 2 (1 + 3 x) + ln 1 + 3 x) + \frac{3}{5} x^{\frac{5}{3}} - \frac{3}{4} x^{\frac{4}{3}} + x - \frac{3}{2} x^{\frac{2}{3}} + 3 3 x - 3 ln 3 x + 1 : x + \frac{3}{5} x^{\frac{5}{3}} - \frac{3}{4} x^{\frac{4}{3}} - \frac{3}{2} x^{\frac{2}{3}} - 3 3 x + \frac{3}{2} (3 x + 1)^{2} - 6

= x + \frac{3}{5} x^{\frac{5}{3}} - \frac{3}{4} x^{\frac{4}{3}} - \frac{3}{2} x^{\frac{2}{3}} - 3 3 x + \frac{3}{2} (3 x + 1)^{2} - 6

解答补常数

= x + \frac{3}{5} x^{\frac{5}{3}} - \frac{3}{4} x^{\frac{4}{3}} - \frac{3}{2} x^{\frac{2}{3}} - 3 3 x + \frac{3}{2} (3 x + 1)^{2} - 6 + C

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