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

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

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integral of ln((4x)^2)

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

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

Solution steps

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

Apply Integration By Parts

: x ln (16 x^{2}) - \int 2 d x

= x ln (16 x^{2}) - \int 2 d x

\int 2 d x = 2 x

= x ln (16 x^{2}) - 2 x

Simplify

ln (16 x^{2}) : 2 (ln (x) + 2 ln (2))

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

Add a constant to the solution

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

\int \frac{10 x}{5 x ^{2} + 1} d x

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

\frac{x ^{2} - 6 x + 1}{x ^{2} + 8}

5 + 7 x

ln (x e^{x} + 9)

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