What is the integral of 4x*ln(x) ?

The integral of 4x*ln(x) is 2x^2ln(x)-x^2+C

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integral of 4x*ln(x)

\int 4 x \cdot ln (x) d x

2 x^{2} ln (x) - x^{2} + C

Solution steps

\int 4 x ln (x) d x

Take the constant out:

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

= 4 \cdot \int x ln (x) d x

Apply Integration By Parts

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

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

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

= 4 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4})

Simplify

4 (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}) : 2 x^{2} ln (x) - x^{2}

= 2 x^{2} ln (x) - x^{2}

Add a constant to the solution

= 2 x^{2} ln (x) - x^{2} + C

f (x) = (\frac{x + 1}{x - 3})^{2}, at x = 7

\frac{\partial}{\partial x} (x^{2} y^{2} cos (x))

\frac{\partial}{\partial x} (ln (1 + x y))

f (x) = \frac{2 x + 3}{3 x + 2}

\frac{\partial}{\partial x} ((e^{x - y}) g oo g)