What is the integral of 2xln^2(x) ?

The integral of 2xln^2(x) is x^2ln^2(x)-x^2ln(x)+(x^2)/2+C

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integral of 2xln^2(x)

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

\int x ln (x) d x = \frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}

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

Simplify

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

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

Add a constant to the solution

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

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