What is the integral of 3x^2ln(x+2) ?

The integral of 3x^2ln(x+2) is x^3ln(x+2)-1/3 ((x+2)^3-9(x+2)^2+36(x+2)-24ln|x+2|)+C

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integral of 3x^2ln(x+2)

\int 3 x^{2} ln (x + 2) d x

x^{3} ln (x + 2) - \frac{1}{3} ((x + 2)^{3} - 9 (x + 2)^{2} + 36 (x + 2) - 24 ln ∣ x + 2 ∣) + C

Solution steps

\int 3 x^{2} ln (x + 2) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int x^{2} ln (x + 2) d x

Apply Integration By Parts

: \frac{1}{3} x^{3} ln (x + 2) - \int \frac{x ^{3}}{3 ( x + 2 )} d x

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

\int \frac{x ^{3}}{3 ( x + 2 )} d x = \frac{1}{9} ((x + 2)^{3} - 9 (x + 2)^{2} + 36 (x + 2) - 24 ln ∣ x + 2 ∣)

= 3 (\frac{1}{3} x^{3} ln (x + 2) - \frac{1}{9} ((x + 2)^{3} - 9 (x + 2)^{2} + 36 (x + 2) - 24 ln ∣ x + 2 ∣))

Simplify

3 (\frac{1}{3} x^{3} ln (x + 2) - \frac{1}{9} ((x + 2)^{3} - 9 (x + 2)^{2} + 36 (x + 2) - 24 ln ∣ x + 2 ∣)) : x^{3} ln (x + 2) - \frac{1}{3} ((x + 2)^{3} - 9 (x + 2)^{2} + 36 (x + 2) - 24 ln ∣ x + 2 ∣)

= x^{3} ln (x + 2) - \frac{1}{3} ((x + 2)^{3} - 9 (x + 2)^{2} + 36 (x + 2) - 24 ln ∣ x + 2 ∣)

Add a constant to the solution

= x^{3} ln (x + 2) - \frac{1}{3} ((x + 2)^{3} - 9 (x + 2)^{2} + 36 (x + 2) - 24 ln ∣ x + 2 ∣) + C

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What is the integral of 3x^2ln(x+2) ?
The integral of 3x^2ln(x+2) is x^3ln(x+2)-1/3 ((x+2)^3-9(x+2)^2+36(x+2)-24ln|x+2|)+C