What is the integral of 4/3 x^{1/2}ln(x) ?

The integral of 4/3 x^{1/2}ln(x) is 4/3 (2/3 x^{3/2}ln(x)-4/9 x^{3/2})+C

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integral of 4/3 x^{1/2}ln(x)

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

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

Solution steps

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

Take the constant out:

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

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

= \frac{4}{3} (\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}}

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

Add a constant to the solution

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

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What is the integral of 4/3 x^{1/2}ln(x) ?
The integral of 4/3 x^{1/2}ln(x) is 4/3 (2/3 x^{3/2}ln(x)-4/9 x^{3/2})+C