Question

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

• 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 $\frac{x^{10}}{2(x^5+x^3+1{)}^2}+C$

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