What is the integral of (ln(sqrt(7x)))/x ?

The integral of (ln(sqrt(7x)))/x is 1/4 ln^2(7x)+C

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integral of (ln(sqrt(7x)))/x

\int \frac{ln ( 7 x )}{x} d x

\frac{1}{4} ln^{2} (7 x) + C

Solution steps

\int \frac{ln ( 7 x )}{x} d x

Apply u-substitution

: \int \frac{ln ( u )}{2 u} d u

= \int \frac{ln ( u )}{2 u} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{ln ( u )}{u} d u

Apply u-substitution

: \int v d v

= \frac{1}{2} \cdot \int v d v

Apply the Power Rule

: \frac{v ^{2}}{2}

= \frac{1}{2} \cdot \frac{v ^{2}}{2}

Substitute back

= \frac{1}{2} \cdot \frac{ln ^{2} ( 7 x )}{2}

Simplify

\frac{1}{2} \cdot \frac{ln ^{2} ( 7 x )}{2} : \frac{1}{4} ln^{2} (7 x)

= \frac{1}{4} ln^{2} (7 x)

Add a constant to the solution

= \frac{1}{4} ln^{2} (7 x) + C

\frac{3 s ^{2}}{( s + 2 ) ( s + 5 )}

\frac{d}{d x} (3 x^{3} - 2 x^{2})

f (x) = (4 x + 3)^{2}

x \to 0 lim (tan^{x} (x))

2 x - 4

What is the integral of (ln(sqrt(7x)))/x ?
The integral of (ln(sqrt(7x)))/x is 1/4 ln^2(7x)+C