Final Answer:
The integral of 7dx divided by x times the quantity x^4 + 8 is equal to (7/8) ln(|x^4 + 8|) + C, where C is the constant of integration.
Step-by-step explanation:
To evaluate the given integral, we start by expressing the denominator as a product of irreducible factors. The denominator x(x^4 + 8) can be factored into x(x^4 + 2^4), and noticing that this resembles the difference of squares pattern, we further simplify it to x(x^2 - 2x + 4)(x^2 + 2x + 4). Now, we can decompose the fraction into partial fractions, A/x + (Bx + C)/(x^2 - 2x + 4) + (Dx + E)/(x^2 + 2x + 4), where A, B, C, D, and E are constants.
After finding the partial fraction decomposition, we integrate each term separately. The integral of A/x is A ln(|x|), the integral of (Bx + C)/(x^2 - 2x + 4) involves arctangent, and the integral of (Dx + E)/(x^2 + 2x + 4) also involves arctangent. After integrating each term, we simplify and combine the terms, and with the help of logarithmic identities, we arrive at the final answer: (7/8) ln(|x^4 + 8|) + C, where C is the constant of integration.
It's crucial to use absolute values in the logarithmic term to account for both positive and negative values of x. The constant of integration, C, represents the arbitrary constant that arises during integration, and it ensures that the indefinite integral encompasses all possible antiderivatives of the given function.