What is the integral of sqrt(t)ln(t) ?

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

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integral of sqrt(t)ln(t)

\int t ln (t) d t

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

Solution steps

\int t ln (t) d t

Apply Integration By Parts

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

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

\int \frac{2}{3} t^{\frac{1}{2}} d t = \frac{4}{9} t^{\frac{3}{2}}

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

Add a constant to the solution

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

\int x^{\frac{5}{4}} d x

t^{2} \frac{d ^{2} y}{d t ^{2}} + 2 t \frac{d y}{d t} - 6 y = 0

\frac{d}{d x} (\frac{4 x ^{2} + 4 x + 4}{x})

\frac{4 s + 3}{s ^{2} + 109 s + 990}

(x + 2)^{2} + (y - 3)^{2} = 37, (- 8, 4)

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