Evaluate: 
dx, a, b > 0.
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
Sol. Let
I = 
dx ... (i)
Then,
I = 
dx
⇒ I = 
dx ... (ii)
Adding (i) and (ii), we get
2I = 
dx
⇒ 2I = 2 π 
dx
⇒ I = π 
Dividing N r and D r by cos 4 x]
⇒ I = π dx
⇒ I = π 
dt, where t = tan x
⇒ I = 
dt
= 
dt
⇒ I = 
(b 2 –a 2 ) 
dt + 
dt
⇒ I = 
I 1 + 
,
where I 1 = 
dt
⇒ I = 
I 1 + 
{tan –1 ∞ – tan –1 0}
⇒ Ι = π 
I 1 +
⇒ I = π 
I 1 + 
... (iii)
To evaluate I 1 , let us consider the integral
tan –1 
= 
dt
⇒ 
tan –1 
= 
⇒ 
tan –1 
= 
– 
dt
⇒ 
tan –1 
= 
+ 2 
dt
⇒ 
tan –1 
= 
+ 2 
dt – 2a 2 
dt
⇒ 
tan –1 
= 
+ 
tan –1 
– 2a 2 
dt
⇒ 2a
2 
dt = 
+ 
tan –1 
⇒ 2a 2 
dt =
⇒ 2a 2 I 1 = 
= 
⇒ I 1 = 
Substituting the value of I 1 in (iii), we get
I = π 
+ 
= 
(a 2 + b 2 )
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