Question

Find the integration of given trigonometric function $\int {\dfrac{{dx}}{{\left( {\operatorname{Sin} x + \operatorname{Cos} x} \right)}}}$
A. $\log \tan \left( {\dfrac{\pi }{8} + \dfrac{x}{2}} \right) + C$
B. $\log \tan \left( {\dfrac{\pi }{8} - \dfrac{x}{2}} \right) + C$
C. $\dfrac{1}{{\sqrt 2 }}\log \tan \left( {\dfrac{\pi }{8} + \dfrac{x}{2}} \right) + C$

Answer 1

First of all we have to multiply by $\dfrac{1}{{\sqrt 2 }}$ in numerator and denominator both to make the formula of $\operatorname{Sin} (a + b)$ that are mentioned below:

Formula used:
$\Rightarrow \operatorname{Sin} (a + b) = \operatorname{Sin} a.\cos b + \cos a.\sin b............................(A)$
Now, we have to take $\operatorname{Sin} (a + b)$form to the numerator then we have to apply the formula of $\int {\cos ecxdx}$ that are mentioned below:
$\Rightarrow \dfrac{1}{{\sin (a + b)}} = \cos ec(a + b)......................(B)$
$\Rightarrow \int {\cos ecxdx = } - \log \left| {\cos ecx + \cot x} \right| + C..........................................(C)$

Complete step by step answer:
Step 1: First of all we have to let the given integration equal to I, that is mentioned below:
$\Rightarrow I = \int {\dfrac{{dx}}{{(\operatorname{Sin} x + \cos x)}}}$
Now, we have to to multiply by $\dfrac{1}{{\sqrt 2 }}$ in numerator and denominator that are mentioned below:
$\Rightarrow I = \int {\dfrac{{\dfrac{1}{{\sqrt 2 }}dx}}{{\dfrac{1}{{\sqrt 2 }}(\operatorname{Sin} x + \cos x)}}} \\\ \Rightarrow I = \dfrac{1}{{\sqrt 2 }}\int {\dfrac{{dx}}{{\sin x.\dfrac{1}{{\sqrt 2 }} + \cos x.\dfrac{1}{{\sqrt 2 }}}}} \\\$
Step 2: We know that $\dfrac{1}{{\sqrt 2 }}$ is equal to $\sin \left( {\dfrac{\pi }{4}} \right)$ and $\cos \left( {\dfrac{\pi }{4}} \right)$ both. So, we have to make the above expression in the formula of $\operatorname{Sin} (a + b)$, that are expressed below:
$\Rightarrow I = \dfrac{1}{{\sqrt 2 }}\int {\dfrac{{dx}}{{\sin x.\cos \left( {\dfrac{\pi }{4}} \right) + \cos x.\sin \left( {\dfrac{\pi }{4}} \right)}}}$
Now, we use the formula (A) on the denominator,
$\Rightarrow I = \dfrac{1}{{\sqrt 2 }}\int {\dfrac{{dx}}{{\sin \left( {x + \dfrac{\pi }{4}} \right)}}}$

Step 3: Now, we convert denominator into numerator by applying formula (B) that are mentioned in the solution hint,
$\Rightarrow I = \dfrac{1}{{\sqrt 2 }}\int {\cos ec\left( {x + \dfrac{\pi }{4}} \right)} dx$
Step 4: Now, we use the formula (C) in the expression mentioned in step 3 to obtain the final value of the given integration.
$\Rightarrow I = \left( {\dfrac{1}{{\sqrt 2 }}} \right) - \log \left| {\cos ec\left( {x + \dfrac{\pi }{4}} \right) + \cot \left( {x + \dfrac{\pi }{4}} \right)} \right| + C$

Hence, with the help of formulas (A) and (B) we have determined the integration of the given function which is as $\int {\dfrac{{dx}}{{\left( {\operatorname{Sin} x + \operatorname{Cos} x} \right)}}}$ $= \left( {\dfrac{1}{{\sqrt 2 }}} \right) - \log \left| {\cos ec\left( {x + \dfrac{\pi }{4}} \right) + \cot \left( {x + \dfrac{\pi }{4}} \right)} \right| + C$. Therefore option (C) is correct.

Note: To find the integration of the function it is necessary that we have to convert the given trigonometric function $\cos ec(a + b)$ in $\sin (a + b)$.
To obtain or convert in the form of formula $\operatorname{Sin} (a + b)$ we have to multiply with $\dfrac{1}{{\sqrt 2 }}$ in the numerator and denominator of the function given.