Evaluate the following integrals:

Given:

$\int \frac{5 x^{4}+12 x^{3}+7 x^{2}}{x^{2}+x} d x$

Now spilt $12 x^{3}$ into $7 x^{3}$ and $5 x^{3}$

$\Rightarrow \int \frac{5 x^{4}+7 x^{3}+5 x^{3}+7 x^{2}}{x^{2}+x} d x$

Now common $5 x^{3}$ from two elements $7 x$ from other two elements,

$\Rightarrow \int \frac{5 x^{2}(x+1)+7 x(x+1)}{x^{2}+x} d x$

$\Rightarrow \frac{\int\left(5 x^{2}+7 x\right)(x+1)}{x(x+1)} d x$

$\Rightarrow \int\left(5 x^{2}+7 x\right) d x$

Now Splitting, we get,

$\Rightarrow \int 5 x^{2} d x+\int 7 x d x$

$\Rightarrow \frac{5 x^{2+1}}{2+1}+\frac{7 x^{1+1}}{1+1}+c$

$\Rightarrow \frac{5 x^{3}}{3}+\frac{7 x^{2}}{2}+c$