The integral
Question:

The integral $\int \frac{3 x^{13}+2 x^{11}}{\left(2 x^{4}+3 x^{2}+1\right)^{4}} d x$ is equal to :

(where $\mathrm{C}$ is a constant of integration)

  1. $\frac{\mathrm{x}^{4}}{\left(2 \mathrm{x}^{4}+3 \mathrm{x}^{2}+1\right)^{3}}+\mathrm{C}$

  2. $\frac{x^{12}}{6\left(2 x^{4}+3 x^{2}+1\right)^{3}}+C$

  3. $\frac{x^{4}}{6\left(2 x^{4}+3 x^{2}+1\right)^{3}}+C$

  4. $\frac{x^{12}}{\left(2 x^{4}+3 x^{2}+1\right)^{3}}+C$


Correct Option: 2

Solution:

$\int \frac{3 x^{13}+2 x^{11}}{\left(2 x^{4}+3 x^{2}+1\right)^{4}} d x$

$\int \frac{\left(\frac{3}{x^{3}}+\frac{2}{x^{5}}\right) d x}{\left(2+\frac{3}{x^{2}}+\frac{1}{x^{4}}\right)^{4}}$

Let $\left(2+\frac{3}{x^{2}}+\frac{1}{x^{4}}\right)=t$

$-\frac{1}{2} \int \frac{\mathrm{dt}}{\mathrm{t}^{4}}=\frac{1}{6 \mathrm{t}^{3}}+\mathrm{C} \Rightarrow \frac{\mathrm{x}^{12}}{6\left(2 \mathrm{x}^{4}+3 \mathrm{x}^{2}+1\right)^{3}}+\mathrm{C}$

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