What is the integral of (tln(t))^2 ?

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

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integral of (tln(t))^2

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

\frac{1}{3} t^{3} ln^{2} (t) - \frac{2}{3} (\frac{1}{3} t^{3} ln (t) - \frac{t ^{3}}{9}) + C

Solution steps

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

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

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

Apply Integration By Parts

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

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

\int \frac{2}{3} t^{2} ln (t) d t = \frac{2}{3} (\frac{1}{3} t^{3} ln (t) - \frac{t ^{3}}{9})

= \frac{1}{3} t^{3} ln^{2} (t) - \frac{2}{3} (\frac{1}{3} t^{3} ln (t) - \frac{t ^{3}}{9})

Add a constant to the solution

= \frac{1}{3} t^{3} ln^{2} (t) - \frac{2}{3} (\frac{1}{3} t^{3} ln (t) - \frac{t ^{3}}{9}) + C

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