Exercise
$\int\left(\frac{x}{3x^2+x+1}\right)dx$
Step-by-step Solution
Learn how to solve problems step by step online. Find the integral int(x/(3x^2+x+1))dx. Rewrite the expression \frac{x}{3x^2+x+1} inside the integral in factored form. Take the constant \frac{1}{3} out of the integral. We can solve the integral \frac{1}{3}\int\frac{x}{\left(x+\frac{1}{6}\right)^2+\frac{11}{36}}dx by applying integration method of trigonometric substitution using the substitution. Now, in order to rewrite d\theta in terms of dx, we need to find the derivative of x. We need to calculate dx, we can do that by deriving the equation above.
Find the integral int(x/(3x^2+x+1))dx
Final answer to the exercise
$\frac{1}{3}\ln\left|\sqrt{\left(x+\frac{1}{6}\right)^2+\frac{11}{36}}\right|+\frac{-\sqrt{11}\arctan\left(\frac{6\left(x+\frac{1}{6}\right)}{\sqrt{11}}\right)}{33}+C_2$