A2 Pure Topic Revision:
Integration
Teacher Led Questions
, 1. Standard Results
A) Easy integra on (from AS pure)
B) Reverse chain rule
C) Trig from formula booklet
D) Rearrange using trig iden es
2. Frac on 2. Integra on by Parts?
A) ln func on Two func ons mul plied together where one
of them “simplifies” when you differen ate
it.
B) Bring up the denominator
Examples:
C) Split the numerator
D) Algebraic division if frac on is improper
E) Par al frac ons
3. Subs tu on
Typically the inner func on is u but usually the ques on
will give you the u-subs tu on.
, Example
1. Identify which process to use for each integration below
(You do not need to solve the integral)
Z Z 2 12 p
(a) x cos x dx (k) x 2x 1 dx u = 2x 1
1
2
Z 2 Z
(b) xe x
dx x
(l) dx u = 2x 3
0 (2x 3)2
Z 6 Z
x 1 p
(c) p dx u=x 2 (m) p sin x dx
3 x 2 x
Z Z
x ln x
(d) p dx u=1 x2 (n) dx
1 x2 x
Z Z
1 p
(e) d✓ (o) x x + 1 dx u=x+1
cos2 2✓
Z Z
1 ⇡
(f) dx (p) x sin x dx
(2x 3)3 0
Z Z
1 x 1
(g) p dx (q)
2
x2 e2x dx
x
0
Z Z
x2 1
(h) dx (r) (x + ex )2 dx
x3 3x + 1
Z Z
1p
(i) ln x dx (s) x2 cos x dx
x
Z 2 Z
2x 1
(j) dx u=x+1 (t) e7x dx
1 (x + 1)2
2
Integration
Teacher Led Questions
, 1. Standard Results
A) Easy integra on (from AS pure)
B) Reverse chain rule
C) Trig from formula booklet
D) Rearrange using trig iden es
2. Frac on 2. Integra on by Parts?
A) ln func on Two func ons mul plied together where one
of them “simplifies” when you differen ate
it.
B) Bring up the denominator
Examples:
C) Split the numerator
D) Algebraic division if frac on is improper
E) Par al frac ons
3. Subs tu on
Typically the inner func on is u but usually the ques on
will give you the u-subs tu on.
, Example
1. Identify which process to use for each integration below
(You do not need to solve the integral)
Z Z 2 12 p
(a) x cos x dx (k) x 2x 1 dx u = 2x 1
1
2
Z 2 Z
(b) xe x
dx x
(l) dx u = 2x 3
0 (2x 3)2
Z 6 Z
x 1 p
(c) p dx u=x 2 (m) p sin x dx
3 x 2 x
Z Z
x ln x
(d) p dx u=1 x2 (n) dx
1 x2 x
Z Z
1 p
(e) d✓ (o) x x + 1 dx u=x+1
cos2 2✓
Z Z
1 ⇡
(f) dx (p) x sin x dx
(2x 3)3 0
Z Z
1 x 1
(g) p dx (q)
2
x2 e2x dx
x
0
Z Z
x2 1
(h) dx (r) (x + ex )2 dx
x3 3x + 1
Z Z
1p
(i) ln x dx (s) x2 cos x dx
x
Z 2 Z
2x 1
(j) dx u=x+1 (t) e7x dx
1 (x + 1)2
2
