2
2
equals
sin
xdxcos
x
∫
(A)
tan x + cot x + C
(B)  tan x – cot x + C
(C)
tan x cot x + C
(D)  tan x – cot 2x + C
7 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
306
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