Integral of \( \displaystyle \frac{3 x}{9 x^{2} + 1} \)

Problem 4.461 · medium

1. \[ \int \frac{3 x}{9 x^{2} + 1}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = 3 \int \frac{x}{9 x^{2} + 1}\, dx \]
   linearity rewriteFactor out the constant 3. Rewrite the numerator to match the derivative of the denominator.✓ Proved
3. \[ = \frac{\int \frac{18 x}{9 x^{2} + 1}\, dx}{6} \]
   linearity simplifyFactor out the constant 1/18. Simplify the coefficient 3/18 to 1/6.✓ Proved
4. \[ = \frac{\ln{\left(9 x^{2} + 1 \right)}}{6} \]
   antiderivativeIntegrate using the substitution rule for log(u).✓ Proved