What is the integral of-2xln(x+3) ?

The integral of-2xln(x+3) is -2(ln(x+3)(1/2 (x+3)^2-3(x+3))-1/4 (x^2-6x-27))+C

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integral of-2xln(x+3)

\int - 2 x ln (x + 3) d x

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

Solution steps

\int - 2 x ln (x + 3) 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 (x + 3) d x

Apply u-substitution

: \int (u - 3) ln (u) d u

= - 2 \cdot \int (u - 3) ln (u) d u

Apply Integration By Parts

: ln (u) (\frac{u ^{2}}{2} - 3 u) - \int \frac{1}{2} (u - 6) d u

= - 2 (ln (u) (\frac{u ^{2}}{2} - 3 u) - \int \frac{1}{2} (u - 6) d u)

\int \frac{1}{2} (u - 6) d u = \frac{1}{2} (\frac{u ^{2}}{2} - 6 u)

= - 2 (ln (u) (\frac{u ^{2}}{2} - 3 u) - \frac{1}{2} (\frac{u ^{2}}{2} - 6 u))

Substitute back

u = x + 3

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of-2xln(x+3) ?
The integral of-2xln(x+3) is -2(ln(x+3)(1/2 (x+3)^2-3(x+3))-1/4 (x^2-6x-27))+C