Specific antiderivatives
Normally the antiderivative is in form of f(x) +C
. But actually we could use some additional information to get the C
and get the function only in terms of x
. And we often call the "additional information" as Initial Conditions
, or f₀(x)
.
Example
We could Integrate the
f'(x)
to getf(x) = 9eˣ + C
.Since
f(8) = 9e⁸ - 8
, we could easily see thatC = -8
So the function under this condition would be:
f(x) = 9eˣ -8
Then we will get the result
f(0) = 9*1 - 8 = 1
.
Example
We could integrate
f'(x)
to getf(x) = x³ - x² + 7x +C
Since
f(6)=200
, so we could substitute6
intof(x)
:f(6) = 6³ - 6² + 7*6 +C = 200
which resultsC = -22
So the function under this condition should be
f(x) = x³ - x² + 7x - 22
And
f(1) = 1³ - 1² + 7*1 - 22 = -15
Example (Separable equations with specific solutions)
It looks confusing, but let's assume
f(x)
asy
sody/dx = 2y
Separate differential equations to get
dy/y = 2dx
Take integral of both sides:
Since
f(1) = 5
, so:And
f(3) = 5e⁴
, which meansm = 5, n = 4
Example
Example
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