Class 12 Integrals · Long Answer3 Marksabsolute_value_branches

Class 12 Maths · Integrals · Long Answer #166

Evaluate \(\int(\sqrt{\cot x}+\sqrt{\tan x})dx\) on any interval where tan x>0. State the sign needed for a real principal-branch answer.

✅ Step-by-Step Solution

Final Answer: s√2 arcsin(sin x−cos x)+C, with s=sign(sin x+cos x).

When tan x>0, sin x and cos x have the same nonzero sign. Let s=sign(sin x+cos x), constant on each such interval.
The sum of positive square roots equals |sin x+cos x|/√(sin x cos x)=s(sin x+cos x)/√(sin x cos x).
Set u=sin x−cos x, du=(sin x+cos x)dx and sin x cos x=(1−u²)/2. The integral becomes s√2∫du/√(1−u²).
Thus the primitive is s√2 arcsin(sin x−cos x)+C. On (0,π/2), s=1; on (π,3π/2), s=−1.

[0.5 marks] When tan x>0, sin x and cos x have the same nonzero sign. Let s=sign(sin x+cos x), constant on each such interval.
[1 mark] The sum of positive square roots equals |sin x+cos x|/√(sin x cos x)=s(sin x+cos x)/√(sin x cos x).
[1 mark] Set u=sin x−cos x, du=(sin x+cos x)dx and sin x cos x=(1−u²)/2. The integral becomes s√2∫du/√(1−u²).
[0.5 marks] Thus the primitive is s√2 arcsin(sin x−cos x)+C. On (0,π/2), s=1; on (π,3π/2), s=−1.

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