Final answer to the problem
$\frac{1}{2}x^2+11\ln\left(x+1\right)+C_0$
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Step-by-step Solution
Specify the solving method
1
Find the integral
$\int\frac{x^2+5+x+6}{x+1}dx$
2
Add the values $5$ and $6$
$\int\frac{11+x^2+x}{x+1}dx$
3
Divide $11+x^2+x$ by $x+1$
$\begin{array}{l}\phantom{\phantom{;}x\phantom{;}+1;}{\phantom{;}x\phantom{;}\phantom{-;x^n}}\\\phantom{;}x\phantom{;}+1\overline{\smash{)}\phantom{;}x^{2}+x\phantom{;}+11\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x\phantom{;}+1;}\underline{-x^{2}-x\phantom{;}\phantom{-;x^n}}\\\phantom{-x^{2}-x\phantom{;};}\phantom{;}11\phantom{;}\phantom{;}\\\end{array}$
4
Resulting polynomial
$\int\left(x+\frac{11}{x+1}\right)dx$
5
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
$\int xdx+\int\frac{11}{x+1}dx$
6
The integral $\int xdx$ results in: $\frac{1}{2}x^2$
$\frac{1}{2}x^2$
7
The integral $\int\frac{11}{x+1}dx$ results in: $11\ln\left(x+1\right)$
$11\ln\left(x+1\right)$
8
Gather the results of all integrals
$\frac{1}{2}x^2+11\ln\left(x+1\right)$
9
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
$\frac{1}{2}x^2+11\ln\left(x+1\right)+C_0$
Final answer to the problem
$\frac{1}{2}x^2+11\ln\left(x+1\right)+C_0$