Final Answer
Step-by-step Solution
Specify the solving method
Rewrite the expression $\frac{1}{3x^2+6x+5}$ inside the integral in factored form
Take the constant $\frac{1}{3}$ out of the integral
We can solve the integral $\int\frac{1}{\frac{2}{3}+\left(x+1\right)^2}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $x+1$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Substituting $u$ and $dx$ in the integral and simplify
Solve the integral by applying the formula $\displaystyle\int\frac{x'}{x^2+a^2}dx=\frac{1}{a}\arctan\left(\frac{x}{a}\right)$
Simplify the expression inside the integral
Replace $u$ with the value that we assigned to it in the beginning: $x+1$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$