A) \[\frac{1}{3}\left[ \log (\sqrt{2}+1)+\frac{\pi }{2\sqrt{2}} \right]\]
B) \[\frac{1}{3}\left[ \log (\sqrt{2}+1)-\frac{\pi }{2\sqrt{2}} \right]\]
C) \[3\left[ \log (\sqrt{2}+1)-\frac{\pi }{2\sqrt{2}} \right]\]
D) \[3\left[ \log (\sqrt{2}+1)+\frac{\pi }{2\sqrt{2}} \right]\]
Correct Answer: A
Solution :
Let \[I=\int_{0}^{\pi /4}{\frac{\cos x}{{{\cos }^{2}}x(1+2{{\sin }^{2}}x)}}\text{ }dx\] \[=\int_{0}^{\pi /4}{\frac{\cos x\,dx}{(1-{{\sin }^{2}}x)(1+2{{\sin }^{2}}x)}}\] \[=\frac{1}{3}\int_{0}^{1/\sqrt{2}}{\left( \frac{1}{1-{{t}^{2}}}+\frac{2}{1+2{{t}^{2}}} \right)}\,dt\] By partial fractions, where \[t=\sin x\] \[=\frac{1}{3}\left[ \frac{1}{2.1}\log \frac{1+t}{1-t}+\frac{2}{\sqrt{2}}{{\tan }^{-1}}t\sqrt{2} \right]_{0}^{1/\sqrt{2}}\] \[=\frac{1}{3}\left[ \frac{1}{2}\log \frac{(\sqrt{2}+1)}{(\sqrt{2}-1)}+\sqrt{2}{{\tan }^{-1}}1 \right]\] \[=\frac{1}{3}\left[ \frac{1}{2}\log {{(\sqrt{2}+1)}^{2}}+\sqrt{2}.\frac{\pi }{4} \right]\]\[=\frac{1}{3}\left[ \log (\sqrt{2}+1)+\frac{\pi }{2\sqrt{2}} \right]\].
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