Key Concepts and Formulas

• Indefinite Integration: Finding a function whose derivative is the given function. The result includes an arbitrary constant of integration, denoted by $c$.
• Substitution Method: A technique to simplify integrals by substituting a part of the integrand with a new variable. If $\int f(g(x))g'(x) \, dx$ is given, substitute $u = g(x)$, then $du = g'(x) \, dx$. The integral becomes $\int f(u) \, du$.
• Power Rule of Integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + c$, where $n \neq -1$.

Step-by-Step Solution

Step 1: Rewrite the integral by factoring out $x^{14}$ from the denominator.

We are given the integral: $f\left( x \right) = \int {{{5{x^8} + 7{x^6}} \over {{{\left( {{x^2} + 1 + 2{x^7}} \right)}^2}}}} \,dx$ Factor out $x^{14}$ from the denominator: $f\left( x \right) = \int {{{5{x^8} + 7{x^6}} \over {{x^{14}}{{\left( {{x^{ - 5}} + {x^{ - 7}} + 2} \right)}^2}}}} \,dx$

Step 2: Simplify the integrand.

Divide the numerator by $x^{14}$: $f\left( x \right) = \int {{{5{x^{ - 6}} + 7{x^{ - 8}}} \over {{{\left( {{x^{ - 5}} + {x^{ - 7}} + 2} \right)}^2}}}} \,dx$

Step 3: Apply substitution.

Let $t = {x^{ - 5}} + {x^{ - 7}} + 2$. Then, differentiate $t$ with respect to $x$: $\frac{dt}{dx} = -5x^{-6} - 7x^{-8}$ $dt = \left( { - 5{x^{ - 6}} - 7{x^{ - 8}}} \right)dx$ $\left( {5{x^{ - 6}} + 7{x^{ - 8}}} \right)dx = - dt$

Step 4: Substitute and integrate.

Substitute $t$ and $dt$ into the integral: $f(x) = \int {{{ - dt} \over {{t^2}}}} = - \int {{t^{ - 2}}dt}$ $f(x) = - \frac{t^{-1}}{-1} + c = \frac{1}{t} + c$

Step 5: Substitute back for $t$.

Substitute $t = {x^{ - 5}} + {x^{ - 7}} + 2$ back into the expression: $f\left( x \right) = {1 \over {{x^{ - 5}} + {x^{ - 7}} + 2}} + c$

Step 6: Simplify the expression.

Multiply the numerator and denominator by $x^7$: $f\left( x \right) = {{{x^7}} \over {{x^2} + 1 + 2{x^7}}} + c$

Step 7: Use the initial condition to find the constant of integration.

We are given $f(0) = 0$. Substitute $x = 0$ into the expression for $f(x)$: $f\left( 0 \right) = {{{{0}^7}} \over {{0^2} + 1 + 2{{\left( 0 \right)}^7}}} + c = \frac{0}{1} + c = 0 + c = 0$ Therefore, $c = 0$.

Step 8: Write the final expression for $f(x)$.

$f\left( x \right) = {{{x^7}} \over {{x^2} + 1 + 2{x^7}}}$

Step 9: Evaluate $f(1)$.

Substitute $x = 1$ into the expression for $f(x)$: $f(1) = {{{1^7}} \over {{1^2} + 1 + 2{{\left( 1 \right)}^7}}} = {1 \over {1 + 1 + 2}} = {1 \over 4}$

Common Mistakes & Tips

• Algebraic Manipulation: Be careful with algebraic manipulations, especially when dealing with negative exponents. A small error can lead to a completely different result.
• Constant of Integration: Don't forget to include the constant of integration, $c$, when evaluating indefinite integrals. Use the initial condition to find its value.
• Substitution: Choosing the right substitution is crucial. In this case, recognizing that the derivative of ${x^{ - 5}} + {x^{ - 7}}$ is related to the numerator of the integrand is key.

Summary

We solved the given integral by first manipulating the integrand to make it suitable for substitution. We let $t = {x^{ - 5}} + {x^{ - 7}} + 2$, which simplified the integral to $\int -t^{-2} dt$. We then integrated, substituted back to get an expression in terms of $x$, and used the given initial condition $f(0) = 0$ to find the constant of integration. Finally, we evaluated $f(1)$, which resulted in $1/4$.

Final Answer

The final answer is \boxed{1/4}, which corresponds to option (D).