3 2  Integration using trigonometric identities
When the integrand involves some trigonometric functions, we use some known identities
to find the integral as illustrated through the following example Example 7 Find (i) 
2
∫cos x dx
 (ii) sin 2 cos 3
x
x dx
∫
 (iii) 
3
∫sin x dx
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MATHEMATICS
Solution
(i)
Recall the identity cos 2x = 2 cos2 x – 1, which gives
cos2x = 1
cos 2
2
x
+
Therefore,      
∫cos2
x dx  = 1 (1+ cos 2 )
2
x dx
∫
= 1
1
cos 2
2
2
dx
x dx
+
∫
∫
= 
1 sin 2
C
2
4
x
x
+
+
(ii)
Recall the identity sin x cos y = 1
2 [sin (x + y) + sin (x – y)]
(Why )
Then   sin 2 cos 3
∫
x
xdx  = 
1
sin 5
sin
2
•




∫
∫
x dx
x dx
= 
1
1 cos 5
cos
C
2
–5
x
x


+
+




= 
1
1
cos 5
cos
C
10
2
–
x
x
+
+
(iii)
From the identity sin 3x = 3 sin x – 4 sin3 x, we find that
sin3x = 3sin
sin 3
4
x –
x
Therefore,      
3
∫sin x dx
 = 3
1
sin
sin 3
4
4
x dx –
x dx
∫
∫
                                      = 
3
1
–
cos
cos 3
C
4
12
x
x
+
+
Alternatively, 
3
2
sin
sin
sin
x dx
x
x dx
=
∫
∫
 = 
(1– cos2
x) sin
x dx
∫
Put cos x = t so that – sin x dx = dt
Therefore,     
3
∫sin x dx
 = 
(
)
1 – t2
dt
−∫
 = 
3
2
C
t3
–
dt
t dt
– t
+
=
+
+
∫
∫
= 
3
1
cos
cos
C
3
–
x
x
+
+
Remark It can be shown using trigonometric identities that both answers are equivalent