What is the integral of ln(x)sqrt(x^5) ?

The integral of ln(x)sqrt(x^5) is 2/7 x^{7/2}ln(x)-4/49 x^{7/2}+C

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integral of ln(x)sqrt(x^5)

\int ln (x) x^{5} d x

\frac{2}{7} x^{\frac{7}{2}} ln (x) - \frac{4}{49} x^{\frac{7}{2}} + C

Solution steps

\int ln (x) x^{5} d x

x^{5} = x^{\frac{5}{2}},

assuming

x \geq 0

= \int ln (x) x^{\frac{5}{2}} d x

Apply Integration By Parts

: \frac{2}{7} x^{\frac{7}{2}} ln (x) - \int \frac{2}{7} x^{\frac{5}{2}} d x

= \frac{2}{7} x^{\frac{7}{2}} ln (x) - \int \frac{2}{7} x^{\frac{5}{2}} d x

\int \frac{2}{7} x^{\frac{5}{2}} d x = \frac{4}{49} x^{\frac{7}{2}}

= \frac{2}{7} x^{\frac{7}{2}} ln (x) - \frac{4}{49} x^{\frac{7}{2}}

Add a constant to the solution

= \frac{2}{7} x^{\frac{7}{2}} ln (x) - \frac{4}{49} x^{\frac{7}{2}} + C

\frac{\partial}{\partial x} (\frac{x - 3 y}{2 x y + 3})

\int \frac{x ^{2} + 10 x + 2}{x ^{3} + 15 x ^{2} + 6 x + 1} d x

\frac{( 3 x ^{3} + 6 )}{x}

\frac{\partial}{\partial x} (3 - 3 x + 4 y - 4 x^{2} y)

What is the integral of ln(x)sqrt(x^5) ?
The integral of ln(x)sqrt(x^5) is 2/7 x^{7/2}ln(x)-4/49 x^{7/2}+C