What is the integral of sqrt(2x)ln(x^3) ?

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

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integral of sqrt(2x)ln(x^3)

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

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

Solution steps

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

2 x = 2 x

= \int 2 x ln (x^{3}) 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^{3}) d x

Apply Integration By Parts

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

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

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

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

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

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

Add a constant to the solution

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

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