Find the most general antiderivative of the function. (Check your answer by differentiation. Use C for the constant of the antiderivative. Remember to use ln |u| where appropriate.)

(x^5+x^3+8x)/x^4

(x^5+x^3+8x)/x^4 = x + 1/x + 8/x^3

so, the antiderivative is

1/2 x^2 + ln|x| - 4/x^2 + C

To find the antiderivative of the function (x^5+x^3+8x)/x^4, we can start by simplifying the expression:

(x^5+x^3+8x)/x^4 = x^1 + x^(-1) + 8x^(-3)

Next, we can find the antiderivative of each term separately. The antiderivative of x^n (where n is any real number except -1) is (x^(n+1))/(n+1) + C.

Applying this rule to each term, we get:

∫ x^1 dx = (x^2)/2 + C₁
∫ x^(-1) dx = ln |x| + C₂
∫ 8x^(-3) dx = 8/(2*(-3)+1) * x^(-2) = -4x^(-2) + C₃

Multiplying each term by the common denominator x^4, we have:

(x^2)/2 * x^4 + ln |x| * x^4 + (-4x^(-2)) * x^4 + C₄

Simplifying further:

(x^6)/2 + x^4 * ln |x| - 4/x^2 + C

Therefore, the most general antiderivative of the function (x^5+x^3+8x)/x^4 is (x^6)/2 + x^4 * ln |x| - 4/x^2 + C.

To check our answer, we can differentiate this function and see if we obtain the original function.

Differentiating the function (x^6)/2 + x^4 * ln |x| - 4/x^2 + C:

d/dx [(x^6)/2 + x^4 * ln |x| - 4/x^2 + C]
= (6/2) x^(6-1) + (4 * ln|x|) x^(4-1) - (4 * (-2)) x^(-2-1) + 0
= 3x^5 + 4x^3 * ln |x| + 8/x^3

As we can see, we obtained the original function!

Thus, we can conclude that the antiderivative (x^6)/2 + x^4 * ln |x| - 4/x^2 + C is correct.