Question

$\int \frac{x^3+1}{\sqrt{x^2+x}}dx$=$\frac{1}{k}[m\sqrt{x^l+x}+nlog|(x+\frac{1}{p})+\sqrt{x^r+x}|]+C$.
Find $k+m+l+n+p+r$ ?

Answer 1

$I=\int \frac{x^3+1}{\sqrt{x^2+x}}dx$
$=\int \frac{x^3+x+1-x}{\sqrt{x^2+x}}dx=\int x\sqrt{x^2+x}dx-\int \frac{x-1}{\sqrt{x^2+x}}dx$
$=\frac{1}{2}[\int (2x+1)\sqrt{x^2+x}dx-\int \sqrt{x^2+x}dx]-\frac{1}{2}\int \frac{2x+1-3}{\sqrt{x^2+x}}dx$
$=\frac{1}{2}[\int (2x+1)\sqrt{x^2+x}dx-\int \sqrt{{(x+\frac{1}{2})}^2-\frac{1}{4}dx}]$
$-\frac{1}{2}[\int \frac{2x+1}{\sqrt{x^2+x}}dx-3\frac{1}{\sqrt{{(x+\frac{1}{2})}^2-\frac{1}{4}}}dx]$
$=\frac{1}{2}[\frac{2{(x^2+x)}^{3/2}}{3}-\frac{x+\frac{1}{2}}{2}\sqrt{x^2+x}+\frac{1}{4}log|(x+\frac{1}{2})+\sqrt{x^2+x}|]$
$=\frac{1}{2}[2\sqrt{x^2+x}-3log|(x+\frac{1}{2})+\sqrt{x^2+x}|]+C$