Find f. (Use C for the constant of the first antiderivative and D for the constant of the second antiderivative.)
f ''(x) = 8 + x3 + x5
Well, if f''(x) = 8 + x^3 + x^5, then we need to find f(x) by integrating twice.
First, we integrate 8 + x^3 + x^5 with respect to x to find f'(x). The integral of 8 with respect to x is 8x, the integral of x^3 with respect to x is (1/4)x^4, and the integral of x^5 with respect to x is (1/6)x^6.
So, f'(x) is equal to 8x + (1/4)x^4 + (1/6)x^6 + C, where C is the constant of integration.
Next, we integrate f'(x) with respect to x to find f(x). The integral of 8x with respect to x is (4/2)x^2, the integral of (1/4)x^4 with respect to x is (1/20)x^5, and the integral of (1/6)x^6 with respect to x is (1/42)x^7.
So, f(x) is equal to (4/2)x^2 + (1/20)x^5 + (1/42)x^7 + Dx + E, where D and E are constants of integration.
Therefore, f(x) = (4/2)x^2 + (1/20)x^5 + (1/42)x^7 + Dx + E.
And that, my friend, is the hilarious answer to your question. Enjoy!
To find f, we need to integrate the given function twice.
First, we integrate f''(x) with respect to x once to find f'(x):
f'(x) = ∫(8 + x^3 + x^5) dx
The antiderivative of 8 with respect to x is 8x.
The antiderivative of x^3 with respect to x is (1/4)x^4.
The antiderivative of x^5 with respect to x is (1/6)x^6.
Therefore, f'(x) = 8x + (1/4)x^4 + (1/6)x^6 + C, where C is a constant.
Next, we integrate f'(x) with respect to x once more to find f(x):
f(x) = ∫(8x + (1/4)x^4 + (1/6)x^6 + C) dx
The antiderivative of 8x with respect to x is (1/2)8x^2 = 4x^2.
The antiderivative of (1/4)x^4 with respect to x is (1/5)(1/4)x^5 = (1/20)x^5.
The antiderivative of (1/6)x^6 with respect to x is (1/7)(1/6)x^7 = (1/42)x^7.
Therefore, f(x) = 4x^2 + (1/20)x^5 + (1/42)x^7 + Cx + D, where C and D are constants.
Hence, we have found f(x) as required.
To find f(x), we need to integrate the given function twice.
Step 1: Find the first antiderivative of f''(x).
Since f''(x) = 8 + x^3 + x^5, we integrate each term separately.
∫(8) dx = 8x + C
∫(x^3) dx = (1/4) x^4 + D
∫(x^5) dx = (1/6) x^6 + E
Note: We used different constants (C, D, E) for each antiderivative since they are separate integration steps.
Step 2: Combine the antiderivatives to find f(x).
Since f(x) is obtained by integrating the second antiderivative, we combine the results from Step 1:
f(x) = 8x + C + (1/4) x^4 + D + (1/6) x^6 + E
Note: We can replace the constants C, D, and E with a single constant (K) since they represent arbitrary constants.
Therefore, the general solution for f(x) is:
f(x) = 8x + (1/4) x^4 + (1/6) x^6 + K
if f ''(x) = 8 + x^3 + x^5 , notice how we write exponents
then f' (x) = 8x +(1/4)x^4 + (1/6)x^6 + C
f(x) = 4x^2 + (1/20)x^5 + (1/42)x^7 + Cx + D