Question

The integral $\int \frac{2x^{12}+5x^9}{(x^5+x^3+1{)}^3}dx$ is equal to:

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

Answer 1

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

$\int \frac{2x^{12}+5x^9}{(x^5+x^3+1{)}^3}dx$

$=\int \frac{2x^{12}+5x^9}{x^{15}(x^{-5}+x^{-2}+1{)}^3}dx$

$=\int \frac{2x^{-3}+5x^{-6}}{(x^{-5}+x^{-2}+1{)}^3}dx$

Let $x^{-5}+x^{-2}+1=t$

$\Rightarrow \int \frac{-dt}{t^3}=\frac{1}{2t^2}+c$

$=\frac{x^{10}}{2(x^5+x^3+1{)}^2}+c$