Integration
Starter:
Find the following derivatives:
d d sec(3x + 2)
(a) 5 sin4 (3x + 2) (b)
dx dx 1 + 2x3
d x d
(c) 4 cot(3x + 2) (d) ln(1 + e3x+2 )
dx dx
, • Recall that integration is the inverse action of di↵erentiation.
Example 0:
These are the standard integrals that you need to learn:
Z Z
(a) xn dx (f) cos(x) dx
Z Z
1
(b) dx (g) sec2 (x) dx
x
Z Z
(c) ex dx (h) cosec2 (x) dx
Z Z
(d) ax dx (i) sec(x) tan(x) dx
Z Z
(e) sin(x) dx (j) cosec(x) cot(x) dx
,Example 1:
A curve C has equation y = 3x 9.
Find the exact area of the finite region bounded between C and the coordinate axes.
, Example 2:
A curve C has equation y = 4 3 cosec2 (x) for 0 < x < ⇡.
Find the exact area of the finite region bounded between C and the x-axis.
Starter:
Find the following derivatives:
d d sec(3x + 2)
(a) 5 sin4 (3x + 2) (b)
dx dx 1 + 2x3
d x d
(c) 4 cot(3x + 2) (d) ln(1 + e3x+2 )
dx dx
, • Recall that integration is the inverse action of di↵erentiation.
Example 0:
These are the standard integrals that you need to learn:
Z Z
(a) xn dx (f) cos(x) dx
Z Z
1
(b) dx (g) sec2 (x) dx
x
Z Z
(c) ex dx (h) cosec2 (x) dx
Z Z
(d) ax dx (i) sec(x) tan(x) dx
Z Z
(e) sin(x) dx (j) cosec(x) cot(x) dx
,Example 1:
A curve C has equation y = 3x 9.
Find the exact area of the finite region bounded between C and the coordinate axes.
, Example 2:
A curve C has equation y = 4 3 cosec2 (x) for 0 < x < ⇡.
Find the exact area of the finite region bounded between C and the x-axis.
