What is the integral of (6x+9)ln(x+3) ?

The integral of (6x+9)ln(x+3) is -3/2 x^2+3x^2ln(x+3)+9xln(x+3)+81/2-27+C

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integral of (6x+9)ln(x+3)

\int (6 x + 9) ln (x + 3) d x

- \frac{3}{2} x^{2} + 3 x^{2} ln (x + 3) + 9 x ln (x + 3) + \frac{81}{2} - 27 + C

Solution steps

\int (6 x + 9) ln (x + 3) d x

Expand

(6 x + 9) ln (x + 3) : 6 x ln (x + 3) + 9 ln (x + 3)

= \int 6 x ln (x + 3) + 9 ln (x + 3) 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 ln (x + 3) d x + \int 9 ln (x + 3) d x

\int 6 x ln (x + 3) d x = 6 (ln (x + 3) (\frac{1}{2} (x + 3)^{2} - 3 (x + 3)) - \frac{1}{4} (x^{2} - 6 x - 27))

\int 9 ln (x + 3) d x = 9 (x ln (x + 3) + 3 ln (x + 3) - x - 3)

= 6 (ln (x + 3) (\frac{1}{2} (x + 3)^{2} - 3 (x + 3)) - \frac{1}{4} (x^{2} - 6 x - 27)) + 9 (x ln (x + 3) + 3 ln (x + 3) - x - 3)

Simplify

6 (ln (x + 3) (\frac{1}{2} (x + 3)^{2} - 3 (x + 3)) - \frac{1}{4} (x^{2} - 6 x - 27)) + 9 (x ln (x + 3) + 3 ln (x + 3) - x - 3) : - \frac{3}{2} x^{2} + 3 x^{2} ln (x + 3) + 9 x ln (x + 3) + \frac{81}{2} - 27

= - \frac{3}{2} x^{2} + 3 x^{2} ln (x + 3) + 9 x ln (x + 3) + \frac{81}{2} - 27

Add a constant to the solution

= - \frac{3}{2} x^{2} + 3 x^{2} ln (x + 3) + 9 x ln (x + 3) + \frac{81}{2} - 27 + C

What is the integral of (6x+9)ln(x+3) ?
The integral of (6x+9)ln(x+3) is -3/2 x^2+3x^2ln(x+3)+9xln(x+3)+81/2-27+C