Question

find f(x) if  f'(x) = 3x^2 - 5x + 3

Accepted Answer

f'(x) 3x^2 - 5x+ 3
f(x) = integral f'(x).
==> f(x) = intg (3x^2 - 5x + 3) dx
                = intg (3x^2)dx - intg (5x) dx + intg 3 dx
                 = 3x^3/3 - 5x^2 /2  + 3x + C
==> f(x) = x^3 - (5/2)x^2 + 3x + C

Answer

We know that F'(x) = f(x), where Int f'(x) dx = f(x).
We'll apply the indefinite integral to the expression of f(x):
Int f'(x) dx = Int (3x^2 - 5x + 3)dx
We'll apply the additive property of indefinite integrals:
Int (3x^2 - 5x + 3)dx = Int 3x^2dx - Int 5xdx + Int 3dx
Int 3x^2dx = 3x^3/3 + C
We'll simplify and we'll get:
Int 3x^2dx = x^3 + C (1)
Int 5xdx = 5x^2/2 + C (2)
Int 3dx = 3x + C (3)
We'll add  (1),(2),(3):
f(x) =  (1)+(2)+(3)
f(x) = x^3 + 5x^2/2 + 3x + C
Note: C+C+C = C (family of constants)

Answer

If f'(x) = 3x^2-5x+3
then  f(x) =I = Int {3x^2-5x+3}dx
I = Int(3x^3) - Int(5x)dx) +Int (3x)dx
I = 3* (1/3+1)x^4-5(1/2) 5x^2 +3x+C , as Int x^n = (1/n+1)x^(n+1).
f(x) = (3/4)x^4 -(5/2)x^2 +3x +C

Answer

Here we are given f'(x) = 3x^2 - 5x + 3. If we integrate f'(x) we get f(x).
Now the integral of x^n is x^(n+1)/ (n+1)
Int [ 3x^2 - 5x + 3 ] = Int [ 3x^2] - Int [ 5x] + Int 3
=> 3x^3/3 - 5x^2 / 2 + 3x
=> 3x^3 - (5/2) x^2 + 3x
Therefore f(x) = 3x^3 - (5/2) x^2 + 3x + C

Math

find f(x) if f'(x) = 3x^2 - 5x + 3

Expert Answers

Unlock This Answer Now

Related Questions

Find the domain f=(5x+4) / (x^2+3x+2)

f(x) = (x^2 + 3x -2)^2 find f'(1)

Given f(x) = k(2+x), find the value of k if f^-1 (-2) = -3

If f(x) = 3x^2 - x + 2, find f(2), f(-2), f(a), f(-a), f(a+1), 2*f(a), f(2a), f(a^2), (f(a))^2 and f(a+h)

Find the inverse function of f(x)=x^3+2?