What is the integral of sqrt(x)ln(13x) ?

The integral of sqrt(x)ln(13x) is 2/3 x^{3/2}ln(13x)-4/9 x^{3/2}+C

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integral of sqrt(x)ln(13x)

\int x ln (13 x) d x

\frac{2}{3} x^{\frac{3}{2}} ln (13 x) - \frac{4}{9} x^{\frac{3}{2}} + C

Solution steps

\int x ln (13 x) d x

Apply Integration By Parts

: \frac{2}{3} x^{\frac{3}{2}} ln (13 x) - \int \frac{2}{3} x^{\frac{1}{2}} d x

= \frac{2}{3} x^{\frac{3}{2}} ln (13 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}}

= \frac{2}{3} x^{\frac{3}{2}} ln (13 x) - \frac{4}{9} x^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} ln (13 x) - \frac{4}{9} x^{\frac{3}{2}} + C

\frac{9}{22 ( 3 s - 2 )}

\int \frac{3 ^{x}}{1 + 3 ^{x}} d x

f (x) = 3 x^{2} + 5 x, at x = - 2

x \to \frac{π}{2} lim (\frac{x}{cos ( x )})

\frac{d y}{d x}, x + 4 y = 1

What is the integral of sqrt(x)ln(13x) ?
The integral of sqrt(x)ln(13x) is 2/3 x^{3/2}ln(13x)-4/9 x^{3/2}+C