A) is independent of a
B) is independent of b
C) is independent of a and b
D) depends on both a and b
Correct Answer: A
Solution :
Let \[I=\,\int_{0}^{\pi /2}{\frac{a+b\,\cos \,x}{{{(b+a\,\cos \,x)}^{2}}}dx}\] \[=\,\int_{0}^{\pi /2}{\frac{\,a\,\cos e{{c}^{2}}\,x+b\,\cot \,x\,\cos ec\,x}{{{(b\,\cos ec\,x+a\,\cot \,x)}^{2}}}dx}\] \[(\because \,\,\text{divide}\,\text{Nr}\,\text{and}\,\text{Dr}\,\text{by}\,{{\sin }^{2}}\,x)\] Let \[b\,\,\cos ec\,x\,+a\,\cot \,x=t\] \[\Rightarrow \] \[(-a\,\cos e{{c}^{2}}x-\cos ec\,x\,\cot \,x)\,dx=dt\] \[\therefore \] \[I=-\,\int_{\infty }^{b}{\frac{1}{{{t}^{2}}}dt=\left( \frac{1}{t} \right)_{\infty }^{b}=\frac{1}{b}}\]
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