The integral 3x^13+2x^11(2x^4+3x^2+1)^4,dx is equal to: (where C is a constant of integrat…

Question

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

Accepted Answer

Divide numerator and denominator by $x^{16}$:
$$I=\int\frac{\frac3{x^3}+\frac2{x^5}}{\left(2+\frac3{x^2}+\frac1{x^4}\right)^4}dx$$

Let $2+\dfrac3{x^2}+\dfrac1{x^4}=t\Rightarrow\left(-\dfrac6{x^3}-\dfrac4{x^5}\right)dx=dt\Rightarrow\left(\dfrac3{x^3}+\dfrac2{x^5}\right)dx=-\dfrac12dt$
$$I=-\frac12\int t^{-4}dt=\frac1{6}t^{-3}+C=\frac{1}{6}\left(2+\frac3{x^2}+\frac1{x^4}\right)^{-3}+C=\frac{x^{12}}{6(2x^4+3x^2+1)^3}+C$$

Solutions · 1