integral of 4/(5-2i)

解答

\int \frac{4}{5 - 2 i} d i

- \frac{4}{29} + i \frac{20}{29} + C

求解步骤

\int \frac{4}{5 - 2 i} d i

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int \frac{1}{5 - 2 i} d i

乘开

\frac{1}{5 - 2 i} : \frac{5}{29} + \frac{2 i}{29}

= 4 \cdot \int \frac{5}{29} + \frac{2 i}{29} d i

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

= 4 (\int \frac{5}{29} d i + \int \frac{2 i}{29} d i)

\int \frac{5}{29} d i = \frac{5}{29} i

\int \frac{2 i}{29} d i = - \frac{1}{29}

= 4 (\frac{5}{29} i - \frac{1}{29})

将

4 (\frac{5}{29} i - \frac{1}{29})

改写成标准复数形式:

- \frac{4}{29} + \frac{20}{29} i

解答补常数

= - \frac{4}{29} + i \frac{20}{29} + C

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