What is the integral of 3ln(x^2) ?

The integral of 3ln(x^2) is 3(2xln(x)-2x)+C

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integral of 3ln(x^2)

\int 3 ln (x^{2}) d x

3 (2 x ln (x) - 2 x) + C

Solution steps

\int 3 ln (x^{2}) d x

Take the constant out:

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

= 3 \cdot \int ln (x^{2}) d x

Apply Integration By Parts

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

= 3 (x ln (x^{2}) - \int 2 d x)

\int 2 d x = 2 x

= 3 (x ln (x^{2}) - 2 x)

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

assuming

x \geq 0

= 3 (2 x ln (x) - 2 x)

Add a constant to the solution

= 3 (2 x ln (x) - 2 x) + C

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