What is the integral of 6x+12xln(8x) ?

The integral of 6x+12xln(8x) is 6x^2ln(8x)+C

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integral of 6x+12xln(8x)

\int 6 x + 12 x ln (8 x) d x

6 x^{2} ln (8 x) + C

Solution steps

\int 6 x + 12 x ln (8 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 6 x d x + \int 12 x ln (8 x) d x

\int 6 x d x = 3 x^{2}

\int 12 x ln (8 x) d x = 12 (\frac{1}{2} x^{2} ln (8 x) - \frac{x ^{2}}{4})

= 3 x^{2} + 12 (\frac{1}{2} x^{2} ln (8 x) - \frac{x ^{2}}{4})

Simplify

3 x^{2} + 12 (\frac{1}{2} x^{2} ln (8 x) - \frac{x ^{2}}{4}) : 6 x^{2} ln (8 x)

= 6 x^{2} ln (8 x)

Add a constant to the solution

= 6 x^{2} ln (8 x) + C

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

(10 y^{2} - x y) d x + x^{2} d y = 0

x \to \infty lim (6 x^{2})

n = 0 \sum \infty \frac{1}{1 + ln ( n )}

\int_{2}^{3} \frac{2}{x ^{2} - 1} d x

What is the integral of 6x+12xln(8x) ?
The integral of 6x+12xln(8x) is 6x^2ln(8x)+C