Evaluate 
dx, |a| > 
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
We have,
I = 
dx
⇒ I = 
dx
⇒ I = 
dx
⇒ I = 
dx
⇒ I = 
dx
⇒ I = 2 
dx
⇒ I = 
dx + 
dx
⇒ I = 
dx + 
d (2a +sin 2x)
⇒ I = 
dt – 
+
C, where t = 2x
⇒ I = I 1 – 
+ C
where I 1 = 
dt
To evaluate I 1 , let us define P given by
P = 
Then, 
= 
⇒ 
= 
⇒ 
= 
⇒ 
= 
+ 
Integrating both sides with respect to, t, we get
P = –2a 
dt + (4a 2 –1) 
dt
⇒ 
= –2a
dt + (4a 2 –1) I 1
⇒ 
= – 
dt + (4a 2 –1) I 1
⇒ 
= – 
+ (4a 2 –1) I 1 ,
where tan 
= u
⇒ 
= – 2 
du + (4a 2 –1) I 1
⇒ 
= – 2 × 
tan –1 
+ (4a 2 –1) I 1
⇒ I 1 = 
. 
+ 
tan –1 
⇒ I 1 = 
+
tan –1 
⇒ I 1 = 
+ 
tan –1
Substituting the value of I 1 in (i), we get
⇒ I 1 = 
+ 
tan –1 
– 
+ C
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