What is the integral of ln(3x^2+4) ?

The integral of ln(3x^2+4) is xln(3x^2+4)+4/(sqrt(3))arctan((sqrt(3))/2 x)-2x+C

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integral of ln(3x^2+4)

\int ln (3 x^{2} + 4) d x

x ln (3 x^{2} + 4) + \frac{4}{3} arctan (\frac{3}{2} x) - 2 x + C

Solution steps

\int ln (3 x^{2} + 4) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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