What is the integral of (ln(5x)+5)/x ?

The integral of (ln(5x)+5)/x is (ln^2(5x))/2+5ln|x|+C

Popular >

integral of (ln(5x)+5)/x

\int \frac{ln ( 5 x ) + 5}{x} d x

\frac{ln ^{2} ( 5 x )}{2} + 5 ln ∣ x ∣ + C

Solution steps

\int \frac{ln ( 5 x ) + 5}{x} d x

Expand

\frac{ln ( 5 x ) + 5}{x} : \frac{ln ( 5 x )}{x} + \frac{5}{x}

= \int \frac{ln ( 5 x )}{x} + \frac{5}{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

= \int \frac{ln ( 5 x )}{x} d x + \int \frac{5}{x} d x

\int \frac{ln ( 5 x )}{x} d x = \frac{ln ^{2} ( 5 x )}{2}

\int \frac{5}{x} d x = 5 ln ∣ x ∣

= \frac{ln ^{2} ( 5 x )}{2} + 5 ln ∣ x ∣

Add a constant to the solution

= \frac{ln ^{2} ( 5 x )}{2} + 5 ln ∣ x ∣ + C

\int_{7}^{9} \frac{1}{x ^{2} - 1} d x

\frac{d}{d x} (x^{3} + x^{2} + \frac{6}{x})

\frac{d}{d x} (\frac{x ^{2} - 1}{x})

\frac{d}{d x} (x^{2} sin (\frac{2 π}{x}))

\int arcsec (x) d x

What is the integral of (ln(5x)+5)/x ?
The integral of (ln(5x)+5)/x is (ln^2(5x))/2+5ln|x|+C