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

The integral of 3x^2ln(x^2+3) is x^3ln(x^2+3)-(2x^3)/3+6x-6sqrt(3)arctan(x/(sqrt(3)))+C

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

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

x^{3} ln (x^{2} + 3) - \frac{2 x ^{3}}{3} + 6 x - 63 arctan (\frac{x}{3}) + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

: \frac{1}{3} x^{3} ln (x^{2} + 3) - \int \frac{2 x ^{4}}{3 ( x ^{2} + 3 )} d x

= 3 (\frac{1}{3} x^{3} ln (x^{2} + 3) - \int \frac{2 x ^{4}}{3 ( x ^{2} + 3 )} d x)

\int \frac{2 x ^{4}}{3 ( x ^{2} + 3 )} d x = \frac{2 x ^{3}}{9} - 2 x + 23 arctan (\frac{x}{3})

= 3 (\frac{1}{3} x^{3} ln (x^{2} + 3) - (\frac{2 x ^{3}}{9} - 2 x + 23 arctan (\frac{x}{3})))

Simplify

3 (\frac{1}{3} x^{3} ln (x^{2} + 3) - (\frac{2 x ^{3}}{9} - 2 x + 23 arctan (\frac{x}{3}))) : x^{3} ln (x^{2} + 3) - \frac{2 x ^{3}}{3} + 6 x - 63 arctan (\frac{x}{3})

= x^{3} ln (x^{2} + 3) - \frac{2 x ^{3}}{3} + 6 x - 63 arctan (\frac{x}{3})

Add a constant to the solution

= x^{3} ln (x^{2} + 3) - \frac{2 x ^{3}}{3} + 6 x - 63 arctan (\frac{x}{3}) + C

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What is the integral of 3x^2ln(x^2+3) ?
The integral of 3x^2ln(x^2+3) is x^3ln(x^2+3)-(2x^3)/3+6x-6sqrt(3)arctan(x/(sqrt(3)))+C