Final answer to the problem
Step-by-step Solution
Specify the solving method
Find the integral
Add the values $5$ and $6$
Divide $x^2+11+x$ by $x+1$
Resulting polynomial
Expand the integral $\int\left(x+\frac{11}{x+1}\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
We can solve the integral $\int\frac{11}{x+1}dx$ 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
The integral $\int xdx$ results in: $\frac{1}{2}x^2$
The integral $\int\frac{11}{u}du$ results in: $11\ln\left(x+1\right)$
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 $C$