integral of ((2+ln(7x)))/(3x)

解答

\int \frac{( 2 + ln ( 7 x ) )}{3 x} d x

\frac{1}{3} (2 ln ∣ x ∣ + \frac{ln ^{2} ( 7 x )}{2}) + C

求解步骤

\int \frac{2 + ln ( 7 x )}{3 x} d x

提出常数:

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

= \frac{1}{3} \cdot \int \frac{2 + ln ( 7 x )}{x} d x

乘开

\frac{2 + ln ( 7 x )}{x} : \frac{2}{x} + \frac{ln ( 7 x )}{x}

= \frac{1}{3} \cdot \int \frac{2}{x} + \frac{ln ( 7 x )}{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

= \frac{1}{3} (\int \frac{2}{x} d x + \int \frac{ln ( 7 x )}{x} d x)

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{ln ( 7 x )}{x} d x = \frac{ln ^{2} ( 7 x )}{2}

= \frac{1}{3} (2 ln ∣ x ∣ + \frac{ln ^{2} ( 7 x )}{2})

解答补常数

= \frac{1}{3} (2 ln ∣ x ∣ + \frac{ln ^{2} ( 7 x )}{2}) + C