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

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

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integral of 2x*ln(x^2)

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

2 x^{2} ln (x) - x^{2} + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Simplify

2 (\frac{1}{2} x^{2} ln (x^{2}) - \frac{x ^{2}}{2}) : 2 x^{2} ln (x) - x^{2}

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

Add a constant to the solution

= 2 x^{2} ln (x) - x^{2} + C

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