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

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

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integral of 2sqrt(x)ln(x)

\int 2 x ln (x) d x

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

Solution steps

\int 2 x ln (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 (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

= 2 (\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}}

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

Add a constant to the solution

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

\frac{d}{d x} (cos (4 e^{- 8 x}))

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

\frac{1}{( 1 + x ^{2} ) ^{2}}

\frac{x + 1}{x + 7}

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

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