BRAMPTON MANOR HARD INTEGRATION Qs and solutions

A2 Pure Topic Revision:
Integra6on

Teacher Led Ques6ons

, 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.

,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

, Discussion
2. Identify which process to use for each integration below
(You do not need to solve the integral)
Z Z
x 2
(a) cos 11x dx (k) dx
x2 4x + 11

Z Z
5x2 x2 x
(b) xe dx (l) dx
x2 3x + 3

Z Z 5
(c) sin x sin(cos x) dx x
(m) p dx u=
0 x+4

Z 4 Z
1
(d) p dx u= (n) 2x(x + 2)5 dx u=
1 1+ x

Z Z
2
2
p p
(e) x x 1 dx u= (o) 3x 2 dx
1


Z Z
1
(f) p dx u= (p) 6x sin(x2 4) dx
x+ x

Z Z
x2
(g) p dx u= (q) 5x cos(5 x2 ) dx
1+x

Z Z
x
(h) dx u= (r) x(x + 2)9 dx u=
1 + 2x

Z Z
x2
(i) dx u= (s) (x + 2)9 dx
1+x

Z Z
1
(j) dx (t) (3x + 2)9 dx
4x + 7




3