Key Concepts and Formulas

• Substitution Method: If we can express an integral in the form $\int f(g(x))g'(x) \, dx$, we can substitute $u = g(x)$, so $du = g'(x) \, dx$, and the integral becomes $\int f(u) \, du$.
• Power Rule for Integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, for $n \neq -1$.

Step-by-Step Solution

Step 1: Rewrite the integral by factoring out $x^{12}$ from the numerator and $x^{16}$ from the denominator.

We are given the integral: $I = \int {{{3{x^{13}} + 2{x^{11}}} \over {{{\left( {2{x^4} + 3{x^2} + 1} \right)}^4}}}} \,dx$ We can rewrite the numerator as $x^{12}(3x + 2x^{-1}) = x^{12}(3x + \frac{2}{x})$ and the denominator as $(x^4)^4 (2 + \frac{3}{x^2} + \frac{1}{x^4})^4 = x^{16}(2 + \frac{3}{x^2} + \frac{1}{x^4})^4$. Thus, $I = \int \frac{x^{12}(3x + \frac{2}{x})}{x^{16}(2 + \frac{3}{x^2} + \frac{1}{x^4})^4} \, dx = \int \frac{3x + \frac{2}{x}}{x^4(2 + \frac{3}{x^2} + \frac{1}{x^4})^4} \, dx = \int \frac{\frac{3}{x^3} + \frac{2}{x^5}}{(2 + \frac{3}{x^2} + \frac{1}{x^4})^4} \, dx$

Step 2: Perform u-substitution.

Let $t = 2 + \frac{3}{x^2} + \frac{1}{x^4}$. Then, we need to find $dt$. $dt = \left( -\frac{6}{x^3} - \frac{4}{x^5} \right) dx = -2 \left( \frac{3}{x^3} + \frac{2}{x^5} \right) dx$ So, $\left( \frac{3}{x^3} + \frac{2}{x^5} \right) dx = -\frac{1}{2} dt$.

Substituting into the integral, we have: $I = \int \frac{-\frac{1}{2} dt}{t^4} = -\frac{1}{2} \int t^{-4} \, dt$

Step 3: Evaluate the integral with respect to $t$.

Using the power rule for integration: $I = -\frac{1}{2} \cdot \frac{t^{-3}}{-3} + C = \frac{1}{6t^3} + C$

Step 4: Substitute back for $t$.

Substituting $t = 2 + \frac{3}{x^2} + \frac{1}{x^4}$ back into the expression: $I = \frac{1}{6 \left( 2 + \frac{3}{x^2} + \frac{1}{x^4} \right)^3} + C = \frac{1}{6 \left( \frac{2x^4 + 3x^2 + 1}{x^4} \right)^3} + C = \frac{1}{6 \frac{(2x^4 + 3x^2 + 1)^3}{x^{12}}} + C = \frac{x^{12}}{6 (2x^4 + 3x^2 + 1)^3} + C$

Common Mistakes & Tips

• Be careful with the signs when differentiating during u-substitution.
• Remember to substitute back to the original variable after integration.
• When dealing with rational functions in integrals, look for ways to simplify the expression before applying integration techniques.

Summary

We simplified the given integral by factoring out appropriate powers of $x$ and then using u-substitution. After evaluating the integral in terms of $t$, we substituted back to obtain the final answer in terms of $x$.

Final Answer

The final answer is $\boxed{\frac{x^{12}}{6(2x^4 + 3x^2 + 1)^3} + C}$, which corresponds to option (A).