Step by step
The question
Question: find y, if dy/dx = x² + 1 and y = 0 when x = 3.
dydx = x2 + 1
1 Find the antiderivative
Go backwards: the antiderivative.
Add 1 to each power, divide by the new power.
And + C.
y = x33 + x + C
Now we have y = x³ over 3 + x + C.
= x33 + 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 = 3 and y = 0.
0 = 333 + 3 + C
Work it out: 12 + C.
= 12 + C
It must equal 0.
12 + C = 0
Take away 12 on both sides: C = −12.
C = 0 − 12 = −12
Careful!
C is not just the y value.
The other terms at x = 3 are not 0 here.
Careful: not C = 0 ✗
3 Write the function
So y = x³ over 3 + x − 12.
y = x33 + x − 12
4 Check
Let's check: its derivative is x² + 1, the slope we were given.
dy/dx = x2 + 1
And at x = 3 it gives 0.
Right!
x = 3: 12 − 12 = 0
Let's look back.
First: find the antiderivative.
Then: find C.
Then: write the function.
Last: check.
Answer
y = x33 + x − 12