Question:

Step 1: Identify the integral expression. The given integral is: $$I = \int \frac{5x^4 + 4x^5}{(x^5 + x + 1)^2} dx$$ Step 2: Manipulate the integrand by dividing both the numerator and denominator by $x^{10}$. To facilitate this, we first rewrite the term $(x^5 + x + 1)$ from the denominator by factoring out $x^5$: $$x^5 + x + 1 = x^5 \left(1 + \frac{x}{x^5} + \frac{1}{x^5}\right) = x^5 \left(1 + x^{-4} + x^{-5}\right)$$ Now, the denominator becomes: $$(x^5 + x + 1)^2 = \left[x^5 \left(1 + x^{-4} + x^{-5}\right)\right]^2 = x^{10} \left(1 + x^{-4} + x^{-5}\right)^2$$ Next, divide the original numerator and denominator by $x^{10}$: $$I = \int \frac{\frac{5x^4 + 4x^5}{x^{10}}}{\frac{(x^5 + x + 1)^2}{x^{10}}} dx = \int \frac{\frac{5x^4}{x^{10}} + \frac{4x^5}{x^{10}}}{\left(\frac{x^5 + x + 1}{x^5}\right)^2} dx$$ Simplify the terms in the numerator and denominator: $$I = \int \frac{5x^{-6} + 4x^{-5}}{\left(1 + x^{-4} + x^{-5}\right)^2} dx$$ Step 3: Perform a substitution to simplify the integral. Let $u$ be the expression inside the parenthesis in the denominator: $$u = 1 + x^{-4} + x^{-5}$$ Now, differentiate $u$ with respect to $x$ to find $du$: $$du = \frac{d}{dx}(1 + x^{-4} + x^{-5}) dx = (-4x^{-5} - 5x^{-6}) dx$$ Factor out $-1$ from $du$: $$du = -(5x^{-6} + 4x^{-5}) dx$$ This means that the numerator of our integrand, $(5x^{-6} + 4x^{-5}) dx$, is equal to $-du$. Step 4: Substitute $u$ and $du$ into the integral and evaluate. The integral transforms into: $$I = \int \frac{-du}{u^2} = -\int u^{-2} du$$ Now, integrate $u^{-2}$ with respect to $u$: $$I = - \left(\frac{u^{-2+1}}{-2+1}\right) + C = - \left(\frac{u^{-1}}{-1}\right) + C = \frac{1}{u} + C$$ Step 5: Substitute back $u = 1 + x^{-4} + x^{-5}$ to express the result in terms of $x$. $$I = \frac{1}{1 + x^{-4} + x^{-5}} + C$$ To simplify the denominator, find a common denominator for the terms $1$, $x^{-4}$, and $x^{-5}$: $$1 + x^{-4} + x^{-5} = 1 + \frac{1}{x^4} + \frac{1}{x^5} = \frac{x^5}{x^5} + \frac{x}{x^5} + \frac{1}{x^5} = \frac{x^5 + x + 1}{x^5}$$ Substitute this simplified expression back into the result for $I$: $$I = \frac{1}{\frac{x^5 + x + 1}{x^5}} + C = \frac{x^5}{x^5 + x + 1} + C$$ Step 6: Compare the result with the given options. The obtained result $I = C + \frac{x^5}{x^5 + x + 1}$ directly matches Option 3. We can also rewrite this expression to see if it matches Option 1: $$C + \frac{x^5}{x^5 + x + 1} = C + \frac{(x^5 + x + 1) - (x + 1)}{x^5 + x + 1}$$ $$= C + \frac{x^5 + x + 1}{x^5 + x + 1} - \frac{x + 1}{x^5 + x + 1}$$ $$= C + 1 - \frac{x + 1}{x^5 + x + 1}$$ Since $C$ is an arbitrary constant of integration, $C+1$ is also an arbitrary constant. Let's denote it as $C'$. So, the result can also be written as $C' - \frac{x + 1}{x^5 + x + 1}$, which matches Option 1. The final answer is $\boxed{C + \frac{x^5}{x^5+x+1}}$ (Option 3) or equivalently $\boxed{C - \frac{x+1}{x^5+x+1}}$ (Option 1).