Question

The integral $\int \frac{3x^{13}+2x^{11}}{(2x^4+3x^2+1{)}^4}dx$ is equal to : (where C is a constant of integration)

• A. $\frac{x^4}{(2x^4+3x^2+1{)}^3}+C$
• B. $\frac{x^{12}}{6(2x^4+3x^2+1{)}^3}+C$
• C. $\frac{x^4}{(2x^4+3x+1{)}^3}+C$
• D. $\frac{x^{12}}{(2x^4+3x+1{)}^3}+C$

Answer 1

Answer: (B)

$\int \frac{3x^{13}+2x^{11}}{(2x^4+3x^2+1{)}^4}dx$
$\int \frac{(\frac{3}{x^3}+\frac{2}{x^5})dx}{(2+\frac{3}{x^2}+\frac{1}{x^4}{)}^4}$
Let $(2\frac{3}{x^2}+\frac{1}{x^4})=t$
$-\frac{1}{2}\int \frac{dt}{t^4}=\frac{1}{6t^3}+C\Rightarrow \frac{x^{1}2}{6(2x^4+3x^2+1{)}^3}+C$