Starter

1.
2.
3.
4.
5.
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
Find the gradient function for each of the
following:
y = x3 + 7
y = x3
y = x3 – 5
y = x3 + 2
y = x3 – 8
What do you notice?
Why do you think this has happened?
Calculus 2
Integration
Calculus 2
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As you saw in the starter
dy/dx = 3x2
is the gradient function for lots of
equations.
How many equations have dy/dx =
3x2?
Curves of the form y = x3 + c, where
c is any number all have dy/dx = 3x2
Integration
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If we are given a gradient function
(dy/dx), integration, is the process of
working backwards from this to find
the equation of the curve.
∫
means “integrate”
Integration
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Is the gradient function enough to
find the equation of the curve?
Suppose dy/dx = 2x
what curve has this gradient
function?
Hint: Think back to the start of differentiation,
what differentiates to give you 2x?
Are there any others?
Differentiation
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What is the general rule for
differentiating xn?
If y = xn then dy/dx = nxn-1
Integration has a similar pattern…..
Integration
If dy/dx = xn then
∫ xn dx = xn+1 + c
n+1
This just means that
you are integrating
‘with respect to x’ as
opposed to any other
letter
This stands for any
constant number
How does this work???
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
The notation can look confusing but
when you put it into practice it gets
easier
You can remember it as:
“add one to the power, divide by the
new power”
Examples

1.
2.
3.
4.
Integrate the following:
x7
x2
5
3x3
Hint:
Remember “add one to the power
and divide by the new power”
What if the equation is more
complicated?
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
Suppose dy/dx = 3x2 + 2x + 5
How do we integrate that?
You just take each part of the
equation at a time to get:
∫ 3x2 + 2x + 5 dx = 3(x3/3) + 2(x2/2) + 5x + c
= x3 + x2 + 5x + c
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Solutions that involve ‘c’ are called
indefinite integrals
INTEGRALS
Finding the value of c
Finding c


Find the general solution to:
dy/dx = 6x2 + 2x – 5
y = 6(x3/3) + 2(x2/2) – 5(x) + c
y = 2x3 + x2 – 5x + c
Finding c
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
Find the equation of the curve with
this gradient function that passes
through the point (1,7)
So we know the equation is of the
form y = 2x3 + x2 – 5x + c
How can we find the value of c?
Finding c
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
7 = 2(13) + (12) - 5(1) + c
7=2+1–5+c
7 = -2 + c
9=c
So the equation of the curve is:
y = 2x3 + x2 – 5x + 9
Summary
To find the equation of a curve:
1. Integrate the gradient function
2. Substitute the coordinates of a
point on the curve to find the value
of c
3. Write the full equation of the curve