Answer :
the final formula for f(x) is: [tex]f(x) = (13/4) * x^4 - 21.25 * x^{0.8} + sin(x) + \pi * x + 84.[/tex]To find a formula for f(x), we need to integrate the given expression for f'(x).
Let's integrate each term separately:
1. For the term 13x^3, we can use the power rule for integration. The power rule states that the integral of x^n is (1/(n+1)) * x^(n+1). Applying this rule, we integrate 13x^3 to get (13/4) * x^4.
2. For the term -(17/(x^.2)), we can use the power rule again. Since the exponent is negative, we can rewrite it as -17 * x^(-0.2). Using the power rule, we integrate -17 * x^(-0.2) to get -17 * (x^(0.8) / 0.8), which simplifies to -21.25 * x^(0.8).
3. For the term cos(x), the integral is simply sin(x).
4. For the term pi, the integral is pi * x.
Now, let's add up all the integrated terms to get the formula for f(x):
[tex]f(x) = (13/4) * x^4 - 21.25 * x^{0.8}+ sin(x) + \pi * x.[/tex]
To find the value of the constant term, we use the given initial condition f(0) = 84. Substituting x = 0 into the formula, we get:
f(0) = (13/4) * 0^4 - 21.25 * 0^(0.8) + sin(0) + pi * 0 = 0 - 0 + 0 + 0 = 0.
Since f(0) = 84, the constant term must be 84.
Therefore, the final formula for f(x) is:
[tex]f(x) = (13/4) * x^4 - 21.25 * x^{0.8} + sin(x) + \pi * x + 84.[/tex]
Know more about the power rule for integration here:
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