What is the integral of sqrt(x)ln(x^9) ?

The integral of sqrt(x)ln(x^9) is 6x^{3/2}ln(x)-4x^{3/2}+C

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integral of sqrt(x)ln(x^9)

\int x ln (x^{9}) d x

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

Solution steps

\int x ln (x^{9}) d x

Simplify

ln (x^{9}) : 9 ln (x)

= \int 9 x ln (x) d x

Take the constant out:

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

= 9 \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

= 9 (\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}}

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

Simplify

9 (\frac{2}{3} x^{\frac{3}{2}} ln (x) - \frac{4}{9} x^{\frac{3}{2}}) : 6 x^{\frac{3}{2}} ln (x) - 4 x^{\frac{3}{2}}

= 6 x^{\frac{3}{2}} ln (x) - 4 x^{\frac{3}{2}}

Add a constant to the solution

= 6 x^{\frac{3}{2}} ln (x) - 4 x^{\frac{3}{2}} + C

\int \frac{x + 1}{4 x + 2} d x

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x e^{- x^{2}}

What is the integral of sqrt(x)ln(x^9) ?
The integral of sqrt(x)ln(x^9) is 6x^{3/2}ln(x)-4x^{3/2}+C