A2 Pure Topic Revision:
Integration
Shadow Questions
, 1. Standard Results
A) Easy integration (from AS pure)
B) Reverse chain rule
C) Trig from formula booklet
D) Rearrange using trig identities
2. Fraction 2. Integration by Parts?
A) ln function Two functions multiplied together where one
of them “simplifies” when you differentiate
it.
B) Bring up the denominator
Examples:
C) Split the numerator
D) Algebraic division if fraction is improper
E) Partial fractions
3. Substitution
Typically the inner function is u but usually the question
will give you the u-substitution.
, 4. Identify which process to use for each integration below
(You do not need to solve the integral)
Z Z p
x2 ln x
(a) dx (k) dx
x3 + 1 x
Z Z
cos(2x + 3) 1
(b) dx (l) 2xe5x dx
sin4 (2x + 3) 0
Z Z
2
(c) x cos(x + 3) dx (m) tan 2x dx
Z Z
2x
(d) dx (n) e2x (e2x + 1)6 dx
cos2 (x2 + 3)
Z ⇡ Z
3x 1
2x 1
(e) sin dx (o) dx u=
0 2 (x 3)2
0
Z ⇡
Z
4 1
(f) dx (p) sin xecos x dx
0 cos2 x
Z 5 Z
2x 1 x3
(g) 3 dx u=x 3 (q) dx
4 (x 3) 2 1+x
Z ✓ ◆2 Z 3
2 4 p
(h) 2 dx (r) x 1 + x2 dx
x 0
Z ⇡ ✓ ◆ Z 2
1
(i) x sin x dx (s) (1 2x) 2 t dx
0 3 1
Z e3 Z 2
1
(j) dx (t) (1 2x) 3 t dx
e x(ln x)2 1
5
Integration
Shadow Questions
, 1. Standard Results
A) Easy integration (from AS pure)
B) Reverse chain rule
C) Trig from formula booklet
D) Rearrange using trig identities
2. Fraction 2. Integration by Parts?
A) ln function Two functions multiplied together where one
of them “simplifies” when you differentiate
it.
B) Bring up the denominator
Examples:
C) Split the numerator
D) Algebraic division if fraction is improper
E) Partial fractions
3. Substitution
Typically the inner function is u but usually the question
will give you the u-substitution.
, 4. Identify which process to use for each integration below
(You do not need to solve the integral)
Z Z p
x2 ln x
(a) dx (k) dx
x3 + 1 x
Z Z
cos(2x + 3) 1
(b) dx (l) 2xe5x dx
sin4 (2x + 3) 0
Z Z
2
(c) x cos(x + 3) dx (m) tan 2x dx
Z Z
2x
(d) dx (n) e2x (e2x + 1)6 dx
cos2 (x2 + 3)
Z ⇡ Z
3x 1
2x 1
(e) sin dx (o) dx u=
0 2 (x 3)2
0
Z ⇡
Z
4 1
(f) dx (p) sin xecos x dx
0 cos2 x
Z 5 Z
2x 1 x3
(g) 3 dx u=x 3 (q) dx
4 (x 3) 2 1+x
Z ✓ ◆2 Z 3
2 4 p
(h) 2 dx (r) x 1 + x2 dx
x 0
Z ⇡ ✓ ◆ Z 2
1
(i) x sin x dx (s) (1 2x) 2 t dx
0 3 1
Z e3 Z 2
1
(j) dx (t) (1 2x) 3 t dx
e x(ln x)2 1
5
