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

The integral of 4sqrt(x)ln(x) is 4(2/3 x^{3/2}ln(x)-4/9 x^{3/2})+C

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integral of 4sqrt(x)ln(x)

\int 4 x ln (x) d x

4 (\frac{2}{3} x^{\frac{3}{2}} ln (x) - \frac{4}{9} x^{\frac{3}{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{2}{3} x^{\frac{3}{2}} ln (x) - \int \frac{2}{3} x^{\frac{1}{2}} d x

= 4 (\frac{2}{3} x^{\frac{3}{2}} ln (x) - \int \frac{2}{3} x^{\frac{1}{2}} d x)

\int \frac{2}{3} x^{\frac{1}{2}} d x = \frac{4}{9} x^{\frac{3}{2}}

= 4 (\frac{2}{3} x^{\frac{3}{2}} ln (x) - \frac{4}{9} x^{\frac{3}{2}})

Add a constant to the solution

= 4 (\frac{2}{3} x^{\frac{3}{2}} ln (x) - \frac{4}{9} x^{\frac{3}{2}}) + C

\int \frac{2 x ^{3}}{( x ^{4} + 5 )} d x

1.12 x^{3} - 6.3 x^{2} - 11.2 x + 1.2

x^{2} e^{x}

\frac{d}{d x} (\frac{i + x j + 2 x k}{5 x ^{2} + 1})

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

What is the integral of 4sqrt(x)ln(x) ?
The integral of 4sqrt(x)ln(x) is 4(2/3 x^{3/2}ln(x)-4/9 x^{3/2})+C