JEE Mains · Maths · STD 12 - 7.1 indefinite integral
The integral \(\int {\sqrt {1 + 2\cot \,x\,\left( {\cos ec\,x + \cot \,x} \right)} \,dx} \) \(\left( {0 < x < \frac{\pi }{2}} \right)\) is equal to ( where \(C\) is a constant of integration)
- A \(2\,\log \,\left| {\sin \frac{x}{2}} \right| + C\)
- B \(4\,\log \,\left| {\sin \frac{x}{2}} \right| + C\)
- C \(2\,\log \,\left| {\cos \frac{x}{2}} \right| + C\)
- D \(4\,\log \,\left| {\cos \frac{x}{2}} \right| + C\)
Answer & Solution
Correct Answer
(A) \(2\,\log \,\left| {\sin \frac{x}{2}} \right| + C\)
Step-by-step Solution
Detailed explanation
Let, \(I=\) \(\int {\sqrt {1 + 2\cot x\csc ecx + 2{{\cot }^2}x} } \cdot dx\) \( \Rightarrow \quad I = \int {\sqrt {\frac{{{{\sin }^2}x + 2\cos x + 2{{\cos }^2}x}}{{{{\sin }^2}x}}} } \cdot dx\) \(\Rightarrow 1=\int \frac{\sqrt{1}+2 \cos x+\cos ^{2} x}{\sin x} \cdot d x\)…
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