What is the integral of (2x+1)ln^2(x) ?

The integral of (2x+1)ln^2(x) is x^2ln^2(x)-x^2ln(x)+(x^2)/2+xln^2(x)-2(xln(x)-x)+C

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integral of (2x+1)ln^2(x)

\int (2 x + 1) ln^{2} (x) d x

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

Solution steps

\int (2 x + 1) ln^{2} (x) d x

Expand

(2 x + 1) ln^{2} (x) : 2 x ln^{2} (x) + ln^{2} (x)

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Add a constant to the solution

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

What is the integral of (2x+1)ln^2(x) ?
The integral of (2x+1)ln^2(x) is x^2ln^2(x)-x^2ln(x)+(x^2)/2+xln^2(x)-2(xln(x)-x)+C