What is the integral of sqrt(2)*x*ln(4x) ?

The integral of sqrt(2)*x*ln(4x) is sqrt(2)(1/2 x^2ln(4x)-(x^2)/4)+C

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integral of sqrt(2)xln(4x)

\int 2 \cdot x \cdot ln (4 x) d x

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

Solution steps

\int 2 x ln (4 x) d x

Take the constant out:

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

= 2 \cdot \int x ln (4 x) d x

Apply Integration By Parts

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

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

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

Add a constant to the solution

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

What is the integral of sqrt(2)*x*ln(4x) ?
The integral of sqrt(2)*x*ln(4x) is sqrt(2)(1/2 x^2ln(4x)-(x^2)/4)+C