UPSEE Solved Model Paper 3

Let I = ∫tan−1λcot−1λ

tanx

tanx+cotx

‌dx ... (i)
I = ∫tan−1λcot−λ

cotx

cotx+tanx

‌dx ... (ii)
(Since tan−1x+cot−1x =

π

2

)
On adding eqs (i) and (ii) we get
2I = ∫tan−1λcot−1λ1‌dx
= [x]tan−λcot−1λ
⇒ 2I = cot−1λ−tan−1λ
⇒ 2I =

π

2

−tan−1λ−tan−1λ
⇒ I =

π

4

−tan−1λ
⇒ −

π

2

< tan−1λ <

π

2

3π

4

>

π

4

−tan−1λ > −

π

4

⇒ −

π

4

< I <

3π

4