Question

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

• A. $\frac{x^{10}}{2{(x^5+x^3+1)}^2}+C$
• B. $\frac{x^2+2x}{(x^5+x^3+1)}+C$
• C. $\log [x^5+x^3+1+\sqrt{2x^{12}+5x^9}+C]$
• D. None of the above

Answer 1

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

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

divide numerator and denominator by $x^{15}$

$=\int \frac{5x^{-6}+2x^{-3}}{{(1+x^{-2}+x^{-5})}^3}dx$
Put $1+x^{-2}+x^{-5}=t.(-2x^{-3}-5x^{-6})dx=dt$
$\therefore I=-\int \frac{dt}{t^2}=\frac{1}{2t^2}=\frac{1}{2{(1+x^{-2}+x^{-5})}^2}$
$=\frac{x^{10}}{2{(x^5+x^3+1)}^2}+C$