Final Answer
Step-by-step Solution
Specify the solving method
Rewrite the expression $\frac{x+3}{x^2+4x+5}$ inside the integral in factored form
We can solve the integral $\int\frac{x+3}{1+\left(x+2\right)^2}dx$ 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 $x+2$ 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
Rewriting $x$ in terms of $u$
Substituting $u$, $dx$ and $x$ in the integral and simplify
Expand the fraction $\frac{1+u}{1+u^2}$ into $2$ simpler fractions with common denominator $1+u^2$
Expand the integral $\int\left(\frac{1}{1+u^2}+\frac{u}{1+u^2}\right)du$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
The integral $\int\frac{1}{1+u^2}du$ results in: $\arctan\left(x+2\right)$
The integral $\int\frac{u}{1+u^2}du$ results in: $\frac{1}{2}\ln\left(1+\left(x+2\right)^2\right)$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$