What is the integral of 2tln(t) ?

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

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integral of 2tln(t)

\int 2 t ln (t) d t

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

Solution steps

\int 2 t ln (t) d t

Take the constant out:

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

= 2 \cdot \int t ln (t) d t

Apply Integration By Parts

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

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

\int \frac{t}{2} d t = \frac{t ^{2}}{4}

= 2 (\frac{1}{2} t^{2} ln (t) - \frac{t ^{2}}{4})

Simplify

2 (\frac{1}{2} t^{2} ln (t) - \frac{t ^{2}}{4}) : t^{2} ln (t) - \frac{t ^{2}}{2}

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

Add a constant to the solution

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

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