Step by step
The question
Question: find y, if dy/dx = 2x + 1 and y = 5 when x = 0.
dydx = 2x + 1
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
y = 2 × x22 + x + C
Now we have y = x² + x + C.
= x2 + x + C
Every antiderivative has + C.
The point will tell us what C is.
+ C: we do not know it yet
2 Find C
Put in x = 0 and y = 5.
5 = 02 + 0 + C
The other terms are 0 at x = 0.
So C = 5.
C = 5
3 Write the function
So y = x² + x + 5.
y = x2 + x + 5
4 Check
Let's check: its derivative is 2x + 1, the slope we were given.
dy/dx = 2x + 1
And at x = 0 it gives 5.
Right!
x = 0: 0 + 5 = 5
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer
y = x2 + x + 5