What is the integral of t^{1/2}ln(t) ?

The integral of t^{1/2}ln(t) is 2/3 t^{3/2}ln(t)-4/9 t^{3/2}+C

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integral of t^{1/2}ln(t)

\int t^{\frac{1}{2}} 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^{\frac{1}{2}} 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

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What is the integral of t^{1/2}ln(t) ?
The integral of t^{1/2}ln(t) is 2/3 t^{3/2}ln(t)-4/9 t^{3/2}+C