What is the integral of (ln(4x^2))/(sqrt(x)) ?

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

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integral of (ln(4x^2))/(sqrt(x))

\int \frac{ln ( 4 x ^{2} )}{x} d x

2 (2 x ln (2 x) - 4 x) + C

Solution steps

\int \frac{ln ( 4 x ^{2} )}{x} d x

Simplify

ln (4 x^{2}) : 2 ln (2 x)

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

Take the constant out:

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

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

Apply Integration By Parts

: 2 x ln (2 x) - \int \frac{2}{x} d x

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

\int \frac{2}{x} d x = 4 x

= 2 (2 x ln (2 x) - 4 x)

Add a constant to the solution

= 2 (2 x ln (2 x) - 4 x) + C

\int x e^{- \frac{x}{100}} d x

\frac{\partial}{\partial y} (9 x y)

e^{- t} \cdot sin (2 t)

\int \frac{x ^{2}}{x ^{2} + 6 x + 10} d x

x \to 0 + lim (x^{14} ln (x))

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