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

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

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integral of (3x^2+y^2)ln(x^2+y^2)

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

ln (y^{2} + x^{2}) (y^{2} x + x^{3}) - \frac{2 x ^{3}}{3} + C

Solution steps

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

Apply Integration By Parts

: ln (y^{2} + x^{2}) (y^{2} x + x^{3}) - \int 2 x^{2} d x

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

\int 2 x^{2} d x = \frac{2 x ^{3}}{3}

= ln (y^{2} + x^{2}) (y^{2} x + x^{3}) - \frac{2 x ^{3}}{3}

Add a constant to the solution

= ln (y^{2} + x^{2}) (y^{2} x + x^{3}) - \frac{2 x ^{3}}{3} + C

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What is the integral of (3x^2+y^2)ln(x^2+y^2) ?
The integral of (3x^2+y^2)ln(x^2+y^2) is ln(y^2+x^2)(y^2x+x^3)-(2x^3)/3+C