What is the integral of t^{3/2}ln(3t) ?

The integral of t^{3/2}ln(3t) is 2/5 t^{5/2}ln(3t)-4/25 t^{5/2}+C

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integral of t^{3/2}ln(3t)

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

\frac{2}{5} t^{\frac{5}{2}} ln (3 t) - \frac{4}{25} t^{\frac{5}{2}} + C

Solution steps

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

Apply Integration By Parts

: \frac{2}{5} t^{\frac{5}{2}} ln (3 t) - \int \frac{2}{5} t^{\frac{3}{2}} d t

= \frac{2}{5} t^{\frac{5}{2}} ln (3 t) - \int \frac{2}{5} t^{\frac{3}{2}} d t

\int \frac{2}{5} t^{\frac{3}{2}} d t = \frac{4}{25} t^{\frac{5}{2}}

= \frac{2}{5} t^{\frac{5}{2}} ln (3 t) - \frac{4}{25} t^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{5} t^{\frac{5}{2}} ln (3 t) - \frac{4}{25} t^{\frac{5}{2}} + C

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What is the integral of t^{3/2}ln(3t) ?
The integral of t^{3/2}ln(3t) is 2/5 t^{5/2}ln(3t)-4/25 t^{5/2}+C