Evaluate the following integrals:

Solution:

Given:

$\int \frac{2 x^{4}+7 x^{3}+6 x^{2}}{x^{2}+2 x} d x$

Take $\mathrm{x}$ is common on both numerator and denominator,

$\Rightarrow \int \frac{x\left(2 x^{3}+7 x^{2}+6 x\right)}{x(x+2)} d x$

$\Rightarrow \int \frac{2 x^{3}+7 x^{2}+6 x}{x+2} d x$

Splitting $7 x^{2}$ into $4 x^{2}$ and $3 x^{2}$

$\Rightarrow \int \frac{2 x^{3}+4 x^{2}+3 x^{2}+6 x}{x+2} d x$

Common the $2 x^{2}$ from first two elements and $3 x$ from next elements,

$\Rightarrow \int \frac{2 x^{2}(x+2)+3 x(x+2)}{x+2} d x$

Now common the $x+2$ from the elements

$\Rightarrow \int \frac{(\mathrm{x}+2)\left(2 \mathrm{x}^{2}+3 \mathrm{x}\right)}{\mathrm{x}+2} \mathrm{dx}$

$\Rightarrow \int\left(2 \mathrm{x}^{2}+3 \mathrm{x}\right) \mathrm{dx}$

Now Splitting, we get,

$\Rightarrow \int 2 x^{2} d x+\int 3 x d x$

Now applying the formula,

$\Rightarrow \frac{2 x^{2+1}}{2+1}+\frac{3 x^{1+1}}{1+1}+c$

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