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

The integral of 3t^2ln(t) is t^3ln(t)-(t^3)/3+C

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integral of 3t^2ln(t)

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

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

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

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

= 3 (\frac{1}{3} t^{3} ln (t) - \frac{t ^{3}}{9})

Simplify

3 (\frac{1}{3} t^{3} ln (t) - \frac{t ^{3}}{9}) : t^{3} ln (t) - \frac{t ^{3}}{3}

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

Add a constant to the solution

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

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What is the integral of 3t^2ln(t) ?
The integral of 3t^2ln(t) is t^3ln(t)-(t^3)/3+C