What is the integral of ((2+ln(7x)))/(3x) ?

The integral of ((2+ln(7x)))/(3x) is 1/3 (2ln|x|+(ln^2(7x))/2)+C

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

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

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

Solution steps

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

Take the constant out:

\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

Expand

\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

Apply the Sum Rule:

\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})

Add a constant to the solution

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

6 x^{2} - 48 x + y^{2} + 5 = 0

x \to (- k + 1) lim (\frac{1}{x - 2})

y^{^{'}} + 8 y = 0

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

\frac{d}{d x} (x^{- \frac{2}{5}} + x^{\frac{2}{5}})

What is the integral of ((2+ln(7x)))/(3x) ?
The integral of ((2+ln(7x)))/(3x) is 1/3 (2ln|x|+(ln^2(7x))/2)+C