Question

$\int \frac{2x^{12}+5x^9}{(x^5+x^3+1{)}^3}dx$ is equal to
(where $C$ is constant of integration)

• A. $\frac{x^5}{2(x^5+x^3+1{)}^2}+C$
• B. $\frac{-x^{10}}{2(x^5+x^3+1{)}^2}+C$
• C. $\frac{-x^5}{(x^5+x^3+1{)}^2}+C$
• D. $\frac{x^{10}}{2(x^5+x^3+1{)}^2}+C$

Answer 1

The correct option is D $\frac{x^{10}}{2(x^5+x^3+1{)}^2}+C$

Let $I=\int \frac{2x^{12}+5x^9}{(x^5+x^3+1{)}^3}dx$
Dividing by $x^{15}$ in numerator and denominator
$\Rightarrow I=\int \frac{\frac{2}{x^3}+\frac{5}{x^6}}{{(1+\frac{1}{x^2}+\frac{1}{x^5})}^3}dx$
Put $1+\frac{1}{x^2}+\frac{1}{x^5}=t$
$\Rightarrow (\frac{-2}{x^3}-\frac{5}{x^6})dx=dt$
$\therefore I=\int \frac{-dt}{t^3}$
$\Rightarrow I=\frac{1}{2t^2}+C$
$\Rightarrow I=\frac{1}{2{(1+\frac{1}{x^2}+\frac{1}{x^5})}^2}+C$
$\therefore I=\frac{x^{10}}{2(x^5+x^3+1{)}^2}+C$