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. $\frac{-x^5}{x^5+x^3+1{)}^2}+C$

Answer 1

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

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

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

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

$=\int \frac{-d(1+\frac{1}{x^2}+\frac{1}{x^5})}{(1+\frac{1}{a^2}+\frac{1}{x^5})}$

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

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