What is the integral of ln(x)sqrt(x^7) ?

The integral of ln(x)sqrt(x^7) is 2/9 x^{9/2}ln(x)-4/81 x^{9/2}+C

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integral of ln(x)sqrt(x^7)

\int ln (x) x^{7} d x

\frac{2}{9} x^{\frac{9}{2}} ln (x) - \frac{4}{81} x^{\frac{9}{2}} + C

Solution steps

\int ln (x) x^{7} d x

x^{7} = x^{\frac{7}{2}},

assuming

x \geq 0

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

Apply Integration By Parts

: \frac{2}{9} x^{\frac{9}{2}} ln (x) - \int \frac{2}{9} x^{\frac{7}{2}} d x

= \frac{2}{9} x^{\frac{9}{2}} ln (x) - \int \frac{2}{9} x^{\frac{7}{2}} d x

\int \frac{2}{9} x^{\frac{7}{2}} d x = \frac{4}{81} x^{\frac{9}{2}}

= \frac{2}{9} x^{\frac{9}{2}} ln (x) - \frac{4}{81} x^{\frac{9}{2}}

Add a constant to the solution

= \frac{2}{9} x^{\frac{9}{2}} ln (x) - \frac{4}{81} x^{\frac{9}{2}} + C

y = - 2 x^{4} + 7 x e^{x}

\int 3 π cos^{3} (5 π x) sin^{7} (5 π x) d x

10 t^{4}

\frac{d y}{d x} - (x + 1) y = x y^{2}

7 t y^{^{'}} + y = 0, y (1) = 1

What is the integral of ln(x)sqrt(x^7) ?
The integral of ln(x)sqrt(x^7) is 2/9 x^{9/2}ln(x)-4/81 x^{9/2}+C