Key Concepts and Formulas

• Indefinite Integration: $\int f'(x) [f(x)]^n dx = \frac{[f(x)]^{n+1}}{n+1} + C$, where $n \neq -1$
• Substitution Method: If $\int f(g(x))g'(x) dx$ is to be evaluated, substitute $u = g(x)$, $du = g'(x) dx$. The integral becomes $\int f(u) du$.
• Fundamental Theorem of Calculus: If $F(x) = \int f(x) dx$, then $F'(x) = f(x)$. Also, $\int_a^b f(x) dx = F(b) - F(a)$.

Step-by-Step Solution

Step 1: Rewrite the integral We are given f(x) = \int {{{5{x^8} + 7{x^6}} \over {{{({x^2} + 1 + 2{x^7})}^2}}}dx We want to manipulate the integrand to make it easier to integrate.

Step 2: Divide numerator and denominator by $x^{14}$ Divide both the numerator and the denominator of the integrand by $x^{14}$: f(x) = \int {{{5{x^8} + 7{x^6}} \over {{{({x^2} + 1 + 2{x^7})}^2}}}dx = \int {{{{5{x^8} + 7{x^6}} \over {{x^{14}}}} \over {{{({x^2} + 1 + 2{x^7})}^2} \over {{x^{14}}}}}dx} $f(x) = \int {{{{5 \over {{x^6}}} + {7 \over {{x^8}}}} \over {{{({{{x^2}} \over {{x^7}}} + {1 \over {{x^7}}} + {{2{x^7}} \over {{x^7}}})}^2}}}dx} = \int {{{{5 \over {{x^6}}} + {7 \over {{x^8}}}} \over {{{({1 \over {{x^5}}} + {1 \over {{x^7}}} + 2)}^2}}}dx}$

Step 3: Apply u-substitution Let $u = 2 + {1 \over {{x^5}}} + {1 \over {{x^7}}}$. Then, ${du \over {dx}} = - {5 \over {{x^6}}} - {7 \over {{x^8}}}$, which implies $du = - \left( {{5 \over {{x^6}}} + {7 \over {{x^8}}}} \right)dx$. Therefore, $\left( {{5 \over {{x^6}}} + {7 \over {{x^8}}}} \right)dx = - du$.

Substituting into the integral, we get $f(x) = \int {{{ - du} \over {{u^2}}}} = - \int {{u^{ - 2}}du}$

Step 4: Integrate with respect to u Integrating, we have $f(x) = - \frac{u^{-1}}{-1} + C = \frac{1}{u} + C$

Step 5: Substitute back for x Substituting back $u = 2 + {1 \over {{x^5}}} + {1 \over {{x^7}}}$, we get $f(x) = {1 \over {2 + {1 \over {{x^5}}} + {1 \over {{x^7}}}}} + C = {{{x^7}} \over {2{x^7} + {x^2} + 1}} + C$

Step 6: Use the initial condition f(0) = 0 to find C We are given that $f(0) = 0$. $f(0) = \frac{0}{2(0) + 0 + 1} + C = 0 + C = 0$ Thus, $C = 0$.

Step 7: Write the expression for f(x) Therefore, $f(x) = {{{x^7}} \over {2{x^7} + {x^2} + 1}}$.

Step 8: Evaluate f(1) We are given that $f(1) = {1 \over K}$. So we need to find $f(1)$. $f(1) = {{{1^7}} \over {2(1)^7 + (1)^2 + 1}} = {1 \over {2 + 1 + 1}} = {1 \over 4}$

Step 9: Solve for K Since $f(1) = {1 \over K}$ and $f(1) = {1 \over 4}$, we have ${1 \over K} = {1 \over 4}$, which implies $K = 4$.

Common Mistakes & Tips

• Be careful when dividing by $x^{14}$. Make sure to divide both the numerator and the denominator.
• Remember to substitute back for $x$ after integrating with respect to $u$.
• Don't forget to add the constant of integration, $C$, and use the initial condition to solve for it.

Summary

We were given an indefinite integral and an initial condition, $f(0) = 0$. First, we manipulated the integrand by dividing both numerator and denominator by $x^{14}$. Then, we used u-substitution to evaluate the integral. After substituting back for $x$, we used the initial condition to find the constant of integration, $C$. Finally, we evaluated $f(1)$ and solved for $K$, where $f(1) = \frac{1}{K}$. The value of K is 4.

Final Answer The final answer is \boxed{4}.