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)

$\int \frac{3x^{13} + 2x^{11}}{(2x^4 + 3x^2 + 1)^4}dx$
$\int \frac{\bigg(\frac{3}{x^3}+\frac{2}{x^5}\bigg)dx}{\bigg(2+\frac{3}{x^2} + \frac{1}{x^4}\bigg)^4}$
Let $\bigg(2\frac{3}{x^2} + \frac{1}{x^4}\bigg) = t$
$-\frac{1}{2} \int \frac{dt}{t^4} \, = \frac{1}{6t^3} + C \, \Rightarrow \, \, \frac{x^12}{6(2x^4 + 3x^2 + 1)^3} + C$