Question

The integral $\int \frac{dx}{(1+\sin x{)}^{1/2}}$ is equal to

• A. $\sqrt{2}\log |\cot \frac{3\pi }{8}-\frac{x}{4}|+c$
• B. $\sqrt{2}\log |\cos ec(\frac{\pi }{4}+\frac{x}{2})-\cot (\frac{\pi }{4}+\frac{x}{2})|+c$
• C. $\sqrt{2}\log |\tan (\frac{\pi }{8}+\frac{x}{4})|+c$
• D. $\sqrt{2}\log |\sec (\frac{\pi }{4}-\frac{x}{2})+\tan (\frac{\pi }{4}-\frac{x}{2})|+c$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $\sqrt{2}\log |\cot \frac{3\pi }{8}-\frac{x}{4}|+c$
B $\sqrt{2}\log |\tan (\frac{\pi }{8}+\frac{x}{4})|+c$
C $\sqrt{2}\log |\sec (\frac{\pi }{4}-\frac{x}{2})+\tan (\frac{\pi }{4}-\frac{x}{2})|+c$
D $\sqrt{2}\log |\cos ec(\frac{\pi }{4}+\frac{x}{2})-\cot (\frac{\pi }{4}+\frac{x}{2})|+c$

Given $\int \frac{dx}{(1+\sin x{)}^{1/2}}$
$\Rightarrow$ $\int \frac{dx}{\left(\cos \right(x/2)+\sin (x/2){)}^{2\times \left(1/2\right)}}$
$\Rightarrow$ $\int \frac{dx}{\left(\right(1/\sqrt{2}\left)\cos \right(x/2)+(1/\sqrt{2}\left)\sin \right(x/2\left)\right)\left(\sqrt{2}\right)}$
$\Rightarrow$ $\int \frac{dx}{\left(\sin \right(\pi /4\left)\cos \right(x/2)+(\cos \left(\pi /4\right)\sin \left(x/2\right)\left)\right(\sqrt{2})}$
$\Rightarrow$ $\int \frac{dx}{\left(\sin \right(\left(\pi /4\right)+\left(x/2\right)\left)\right(\sqrt{2})}$
$\Rightarrow$ $\int \frac{1}{\sqrt{2}}\cos ec\left(\right(x/2)+(\pi /4\left)\right)dx$
Since $\int \cos ec\left(x\right)dx=log|\cos ec\left(x\right)-\cot \left(x\right)|$
Hence $\left(\sqrt{2}\right)log|\cos ec\left(\right(x/2)+(\pi /4\left)\right)-\cot \left(\right(x/2)+(\pi /4\left)\right)|+c$ ie option $B$ is the answer.