If f is the antiderivative of g(x)=(x^3)/(1+x^5) such that f(1)=0, then f(5)=___.
Unless you meant 1 + x^4 in the denominator or x^4 in the numerator, that integral is a very difficult one to do. Are you sure you typed g(x) correctly?
There is a recursion formula for that integral in my Table of Integrals, but it leads to two new integrals that look even worse.
To find the value of f(5), we need to evaluate the antiderivative of g(x) at x = 5. Here's how you can do it:
First, find the antiderivative of g(x). The function g(x) can be expressed as:
g(x) = (x^3)/(1+x^5)
To find the antiderivative of g(x), you can use the substitution method. Let u = 1 + x^5. Then, differentiate both sides of this equation with respect to x to find du/dx:
du/dx = 5x^4
Rearrange the equation to solve for dx:
dx = du / (5x^4)
Now substitute the values of u and dx into the integral:
∫ (x^3)/(1+x^5) dx = ∫ (1/u) du/ (5x^4)
Split the integral into two parts:
∫ du / (5x^4u) = (1/5) ∫ du/u - (1/5) ∫ du/x^4
Integrate each part separately:
(1/5) ln|u| - (1/5) (1/3) x^-3 + C
Replace u with 1 + x^5:
(1/5) ln|1 + x^5| - (1/5) (1/3) x^-3 + C
Since f(1) = 0, we can determine the value of C. Substitute x = 1 into the expression:
(1/5) ln|1 + 1^5| - (1/5) (1/3) 1^-3 + C = 0
Simplifying the equation:
(1/5) ln(2) - (1/15) + C = 0
(1/5) ln(2) - (1/15) = C
Now we have our function f(x):
f(x) = (1/5) ln|1 + x^5| - (1/5) (1/3) x^-3 + (1/5) ln(2) - (1/15)
To find f(5), substitute x = 5 into the expression:
f(5) = (1/5) ln|1 + 5^5| - (1/5) (1/3) 5^-3 + (1/5) ln(2) - (1/15)
Evaluating the expression will give you the value of f(5).