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

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

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integral of ln(x^2+3x)

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

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

Solution steps

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

Multiply by the conjugate of

x^{2} + 3 x : \frac{x ^{4} - 9 x ^{2}}{x ^{2} - 3 x}

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

\frac{x ^{4} - 9 x ^{2}}{x ^{2} - 3 x} = x (x + 3)

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

Complete the square

x (x + 3) : (x + \frac{3}{2})^{2} - \frac{9}{4}

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

Apply u-substitution

: \int ln (u^{2} - \frac{9}{4}) d u

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

Apply Integration By Parts

: u ln (u^{2} - \frac{9}{4}) - \int \frac{8 u ^{2}}{4 u ^{2} - 9} d u

= u ln (u^{2} - \frac{9}{4}) - \int \frac{8 u ^{2}}{4 u ^{2} - 9} d u

\int \frac{8 u ^{2}}{4 u ^{2} - 9} d u = - \frac{3}{2} ln \frac{2}{3} u + 1 + \frac{3}{2} ln \frac{2}{3} u - 1 + 2 u

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

Substitute back

u = x + \frac{3}{2}

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

Simplify

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

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

Add a constant to the solution

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

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