Question

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

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

Answer 1

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

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

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

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

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

$-2x^{-3}-5x^{-6}dx=dt$

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

$\Rightarrow \frac{1}{2t^2}+c\Rightarrow \frac{1}{2(1+x^{-2}+x^{-5}{)}^2}+c$

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