What is the integral of 12x^2ln(x) ?

The integral of 12x^2ln(x) is 12(1/3 x^3ln(x)-(x^3)/9)+C

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integral of 12x^2ln(x)

\int 12 x^{2} ln (x) d x

12 (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}) + C

Solution steps

\int 12 x^{2} ln (x) d x

Take the constant out:

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

= 12 \cdot \int x^{2} ln (x) d x

Apply Integration By Parts

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

= 12 (\frac{1}{3} x^{3} ln (x) - \int \frac{x ^{2}}{3} d x)

\int \frac{x ^{2}}{3} d x = \frac{x ^{3}}{9}

= 12 (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9})

Add a constant to the solution

= 12 (\frac{1}{3} x^{3} ln (x) - \frac{x ^{3}}{9}) + C

arccot (15 y)

\frac{d}{d x} (\frac{3 x}{2})

x^{2} y^{^{'}} + 2 x y = 13 cos^{2} (x)

y^{^{'}} + 3 x^{2} y = 6 x^{2}

\int - cot (2 x) d x

What is the integral of 12x^2ln(x) ?
The integral of 12x^2ln(x) is 12(1/3 x^3ln(x)-(x^3)/9)+C