Find the general antiderivative of f (x) and check your answer by differentiating.
f(x) = x^5 + 9
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The rules of the derivative of x^n is nx^(n-1).
The anti-derivative is the reverse process: ∫x^ndx = (1/(n+1))x^(n+1) + C
To find the antiderivative of f(x), we can apply the power rule for integration. The power rule states that for any constant a, ∫x^a dx = (x^(a+1))/(a+1) + C, where C is the constant of integration.
Applying the power rule to f(x) = x^5 + 9, we can integrate each term separately:
∫(x^5) dx = (x^(5+1))/(5+1) + C1 = (x^6)/6 + C1
∫9 dx = 9x + C2
Combining these terms, we find the general antiderivative of f(x):
∫f(x) dx = (x^6)/6 + 9x + C
To check our answer, we can differentiate this result and see if it matches the original function f(x).
Taking the derivative of (x^6)/6 + 9x, we use the power rule for differentiation, which states that d/dx(x^a) = a*x^(a-1):
d/dx[(x^6)/6 + 9x] = (6*x^(6-1))/6 + 9 = x^5 + 9
We can see that the derivative matches the original function f(x) = x^5 + 9. Therefore, our answer for the antiderivative is correct.