how to find f(x) from f'(x)?

Question

how do you find a general quadratic function g(x) whose derivative g'(x)=5x+7?

Answer 1

You integrate g'(x) to get g(x). You typically learn about integration, which is the mathematical inverse of differentiation, the semester or quarter after you learn about differentiation. Integration, like differentiation, is linear. That is, for any constant c and any functions h(x) and k(x),

∫ ch(x) dx = c∫ h(x) dx and
∫ h(x) + k(x) dx = ∫ h(x) dx + ∫ k(x) dx

The inverse relationship between integration and differentiation can be expressed as

d(∫ h(x) dx) / dx = h(x)
and
∫ h'(x) dx = h(x) + C, where C is the constant of integration. C can be any constant, but in many applications some other aspect of the problem pins C down to a specific value.

There is also the analog of the power rule for integration, similar to the rule
d(x^n) / dx = n*x^(n-1), which goes

∫ x^n dx = x^(n+1) / (n+1) + C

These can be combined to find
∫ g'(x) dx
= ∫ (5x + 7) dx
= ∫ 5x dx + ∫ 7 dx
= 5∫ x dx + 7∫ 1 dx
= 5(x^2 / 2) + C1 + 7(x + C2)
= 5/2x^2 + 7x + C, where C is arbitrary, pending further information to pin it down. This particular problem lacks any restrictions on C. If we define
g(x) = 5/2x^2 + 7x + C,
then g'(x) = 5x + 7 and our condition for g(x) is met.

Answer 2

You integrate it.

Integral of 5x + 7 = 2.5x^2 +7x + c


which a reversal of the derivative process.

Answer 3

integrate.
This will leave you with an unknown constant. So the question should give you a value of x where the value of y is known (or vice versa), you can calculate C from this.

g(x) = ∫ g'(x) dx = ∫(5x + 7) dx = (5/2)x² + 7x + C

Answer 4

It's simply the integral of that function:
5/2x² + 7x + C

Answer 5

by integration

g'(x) = 5x + 7

integrating both sides

g(x) = 5x^2/2 + 7x + c

Answer 6

just have to integrate f'(x)

so the integral of f'(x)=f(x)
integral of 5x+7=
(5x^2)/2 +7x
f(x)=(5x^2)/2 +7x integrate f(x) to see you get 5x+7
i mean g(x)

Answer 7

g(x) = 5x^2/2 +7x + c

Answer 8

(5x/2)^2 + 7x + c

Answer 9

u get it as easy as pie
use integral calculus of power

therfore g(x)=5/2x^2+7x+c

Answer 10

Everyone who said "integrate" is right -- Fundamental Theorem of Calculus.