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

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

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

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

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

Solution steps

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

Expand

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

= \int x^{2} ln (4 x^{2}) + ln (4 x^{2}) 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 x^{2} ln (4 x^{2}) d x + \int ln (4 x^{2}) d x

\int x^{2} ln (4 x^{2}) d x = \frac{2}{3} x^{3} ln (2 x) - \frac{2 x ^{3}}{9}

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

= \frac{2}{3} x^{3} ln (2 x) - \frac{2 x ^{3}}{9} + 2 x ln (2 x) - 2 x

Add a constant to the solution

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

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