How do you integrate #xln(x^2)#?

To integrate ( x \ln(x^2) ), we can use integration by parts. Let ( u = \ln(x^2) ) and ( dv = x , dx ). Then differentiate ( u ) to find ( du ), and integrate ( dv ) to find ( v ). After that, apply the integration by parts formula:

[ \int u , dv = uv - \int v , du ]

Here are the steps:

1. Let ( u = \ln(x^2) ), so ( du = \frac{1}{x^2} , dx ).
2. Let ( dv = x , dx ), so ( v = \frac{x^2}{2} ).

Now apply the integration by parts formula:

[ \int x \ln(x^2) , dx = \frac{x^2 \ln(x^2)}{2} - \int \frac{x^2}{2} \cdot \frac{1}{x^2} , dx ]

Simplify the integral on the right-hand side:

[ \frac{x^2 \ln(x^2)}{2} - \int \frac{1}{2} , dx ]

Integrate the remaining term:

[ \frac{x^2 \ln(x^2)}{2} - \frac{x}{2} + C ]

So the integral of ( x \ln(x^2) ) is ( \frac{x^2 \ln(x^2)}{2} - \frac{x}{2} + C ), where ( C ) is the constant of integration.