Integration
Starter:
✓ ◆10
1
(a) Find the first four terms, in ascending powers of x, for the binomial expansion of 2+ x .
2
(b) Use your expansion to estimate the value of 2.110 .
, • Notation:
Z
f (x) dx
• Integration is the opposite (inverse action) of Di↵erentiation.
Example 0:
If f 0 (x) = 2x, then what is f (x)?
, • You must put +C whenever you integrate.
Z
xn+1
• xn dx = +C
n+1
“Increase the power by one, then divide by the new power”
Example 1:
Find the following integrals:
Z Z
p
(a) 3x2 dx (d) 2 x dx
Z Z
(b) 4x6 dx (e) 2021 dx
Z Z
3 1
(c) dx (f) dx
x2 x
, • Expand and simplify first, as you would for di↵erentiation.
Example 2:
Find the following integrals:
Z Z
(a) (x2 ⇥ x3 + x4 ) dx (b) (ax2 + bx + c) dx
Z Z ✓ ◆
(2x 3)2 p 6
(c) dx (d) 2x x p dx
x4 4
x
Z Z
2x3 ⇡x2 + 7 (4x + 3)(4x 3)
(e) dx (f) p dx
5x2 12 x
Starter:
✓ ◆10
1
(a) Find the first four terms, in ascending powers of x, for the binomial expansion of 2+ x .
2
(b) Use your expansion to estimate the value of 2.110 .
, • Notation:
Z
f (x) dx
• Integration is the opposite (inverse action) of Di↵erentiation.
Example 0:
If f 0 (x) = 2x, then what is f (x)?
, • You must put +C whenever you integrate.
Z
xn+1
• xn dx = +C
n+1
“Increase the power by one, then divide by the new power”
Example 1:
Find the following integrals:
Z Z
p
(a) 3x2 dx (d) 2 x dx
Z Z
(b) 4x6 dx (e) 2021 dx
Z Z
3 1
(c) dx (f) dx
x2 x
, • Expand and simplify first, as you would for di↵erentiation.
Example 2:
Find the following integrals:
Z Z
(a) (x2 ⇥ x3 + x4 ) dx (b) (ax2 + bx + c) dx
Z Z ✓ ◆
(2x 3)2 p 6
(c) dx (d) 2x x p dx
x4 4
x
Z Z
2x3 ⇡x2 + 7 (4x + 3)(4x 3)
(e) dx (f) p dx
5x2 12 x
