What is the integral of 9x^2ln(x) ?

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

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integral of 9x^2ln(x)

\int 9 x^{2} ln (x) d x

3 x^{3} ln (x) - x^{3} + C

Solution steps

\int 9 x^{2} ln (x) d x

Take the constant out:

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

= 9 \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

= 9 (\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}

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

Simplify

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

= 3 x^{3} ln (x) - x^{3}

Add a constant to the solution

= 3 x^{3} ln (x) - x^{3} + C

x \to 6 + lim (ln (x^{2} - 36))

y^{^{'}} + x y = x

y = 7 e^{1 - x^{2}}, at

\frac{d}{d x} ((1 - \frac{x}{2})^{e^{x^{2}}})

x \frac{d y}{d x} + (3 x + 1) y = e^{3 x}

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