Consider the function f(x)=x^4 + 8sqrtx
Let F(x) be the antiderivative of f(x) with F(1)=-4.
Then F(x)=_____
To find the antiderivative F(x) of the function f(x), we can use the power rule for integration.
The power rule states that if we have a function of the form x^n, then its antiderivative will be (1/(n+1))x^(n+1) + C, where C is the constant of integration.
Applying the power rule to each term of the given function f(x), we get:
F(x) = (1/(4+1))x^(4+1) + (1/(2+1))8x^(1+1) + C
= (1/5)x^5 + (1/3)8x^2 + C
= (1/5)x^5 + (8/3)x^2 + C
Now, to determine the value of C, we can use the fact that F(1) = -4. Substituting x = 1 into the expression for F(x), we have:
-4 = (1/5)(1)^5 + (8/3)(1)^2 + C
-4 = 1/5 + 8/3 + C
To solve for C, we combine the fractions and simplify:
-4 = 3/15 + 40/15 + C
-4 = (3 + 40)/15 + C
-4 = 43/15 + C
To isolate C, we subtract 43/15 from both sides:
C = -4 - 43/15
C = (-60 - 43)/15
C = -103/15
Therefore, the antiderivative F(x) with F(1) = -4 is:
F(x) = (1/5)x^5 + (8/3)x^2 - (103/15)