difinite integral f(x)=8-4� cube root-3(x) from x=0 to x=8
To evaluate the definite integral of the function f(x) = 8 - 4∛(-3x) from x = 0 to x = 8, follow these steps:
Step 1: Find the antiderivative of the function f(x).
To find the antiderivative of f(x), integrate each term separately.
- The integral of 8 with respect to x is 8x.
- The integral of -4 is -4x.
- The integral of ∛(-3x) can be computed by a variable substitution u = -3x:
Let u = -3x
Then, du/dx = -3
Solving for dx: dx = du / (-3)
Now, substitute the variables back into the integral:
∫ ∛(-3x) dx = ∫ ∛u * (du / -3) = -∫∛u du / 3.
To evaluate ∫∛u du, let v = ∛u:
Then, dv/du = 1 / (3∛(u^2))
Solving for du: du = 3∛(u^2) dv.
Substituting these variables back into the integral:
-∫∛u du / 3 = -∫v (3∛(u^2) dv) / 3
= -∫v ∛(u^2) dv
= -∫v * v^2 dv
= -∫v^3 dv
= - (1/4) * v^4
= - (1/4) * (∛u)^4
= - (1/4) * (u^(4/3))
= - (1/4) * ((-3x)^(4/3))
Combining the integral of each term, we have:
∫[0 to 8] f(x) dx = (8x - 4x - (1/4) * ((-3x)^(4/3)))
Step 2: Evaluate the definite integral by substituting the upper and lower limits.
To find the value of the definite integral, substitute x = 8 and x = 0 into the equation obtained in step 1:
(8(8) - 4(8) - (1/4) * ((-3(8))^(4/3))) - (8(0) - 4(0) - (1/4) * ((-3(0))^(4/3)))
= (64 - 32 - (1/4) * ((-24)^(4/3))) - (0 - 0 - (1/4) * ((-3(0))^(4/3)))
= 32 - (1/4) * ((-24)^(4/3))
= 32 - (1/4) * (64)
= 32 - 16
= 16
Therefore, the definite integral of f(x) = 8 - 4∛(-3x) from x = 0 to x = 8 is equal to 16.