What is the integral of 4xln(5x) ?

The integral of 4xln(5x) is 2x^2ln(5x)-x^2+C

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integral of 4xln(5x)

\int 4 x ln (5 x) d x

2 x^{2} ln (5 x) - x^{2} + C

Solution steps

\int 4 x ln (5 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 (5 x) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} ln (5 x) - \int \frac{x}{2} d x

= 4 (\frac{1}{2} x^{2} ln (5 x) - \int \frac{x}{2} d x)

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

= 4 (\frac{1}{2} x^{2} ln (5 x) - \frac{x ^{2}}{4})

Simplify

4 (\frac{1}{2} x^{2} ln (5 x) - \frac{x ^{2}}{4}) : 2 x^{2} ln (5 x) - x^{2}

= 2 x^{2} ln (5 x) - x^{2}

Add a constant to the solution

= 2 x^{2} ln (5 x) - x^{2} + C

y^{^{'}} = 4 y^{2} + x y^{2}, y (0) = 1

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