Question

Find the value of the integral $\int {\dfrac{1}{{1 + \cos ax}}dx} $. Choose the correct option as the answer from the options given below.
1) $\cot \dfrac{{ax}}{2} + c$
2) $\dfrac{1}{a}\tan \dfrac{{ax}}{2} + c$
3) $\dfrac{1}{a}\left( {\operatorname{cosec} ax - \cot ax} \right) + c$
4) $\dfrac{1}{a}\left( {\operatorname{cosec} ax + \cot ax} \right) + c$

Answer 1

Hint: Given a function to integrate. We cannot integrate the given function directly so we need to solve it and make it a suitable form to integrate. First, we will use the $$\cos 2x$$ formula to simplify the given function. And then transfer it to $\sec $ from that it is easy to integrate.

Complete step-by-step answer:
Given a function to integrate.
Let us name them as
$f\left( x \right) = \dfrac{1}{{1 + \cos ax}}$
Let us simplify it.
We have,
$\cos 2x = 2{\cos ^2}x - 1$
From that we get,
$ \Rightarrow 1 + \cos 2x = 2{\cos ^2}x$.
Let us replace $x$ with $\dfrac{{ax}}{2}$ .
We get,
$1 + \cos ax = 2{\cos ^2}\dfrac{{ax}}{2}$.
Let us substitute this value in the given function.
$f\left( x \right) = \dfrac{1}{{2{{\cos }^2}\dfrac{{ax}}{2}}}$
We know that $\sec $ is a reciprocal function of $\cos $
Therefore,
$f\left( x \right) = \dfrac{1}{2}{\sec ^2}\dfrac{{ax}}{2}$.
We need, $\int {f\left( x \right).dx} $
$ \Rightarrow \int {f\left( x \right).dx} = \int {\dfrac{1}{2}{{\sec }^2}\dfrac{{ax}}{2}dx} $ ,
Let us take the constant out of the integral.
$ \Rightarrow \int {f\left( x \right).dx} = \dfrac{1}{2}\int {{{\sec }^2}\dfrac{{ax}}{2}dx} $ ,
Now let us equate try to equate the variable inside the function and $dx$
To do that we need to multiply and divide $\dfrac{a}{2}$ inside the integral.
We get,
$$ \Rightarrow \int {f\left( x \right).dx} = \dfrac{1}{2}\int {{{\sec }^2}\dfrac{{ax}}{2}.\dfrac{a}{2}.\dfrac{2}{a}.dx} $$
Let us take $\dfrac{2}{a}$out of the integral,
We get,
$$ \Rightarrow \int {f\left( x \right).dx} = \dfrac{1}{2}.\dfrac{2}{a}\int {{{\sec }^2}\dfrac{{ax}}{2}.\dfrac{a}{2}.dx} $$ ,
Let us cancel the outside two and send the $\dfrac{a}{2}$ inside the $dx$ .
We get,
$$ \Rightarrow \int {f\left( x \right).dx} = \dfrac{1}{a}.\int {{{\sec }^2}\dfrac{{ax}}{2}.d\left( {\dfrac{{ax}}{2}} \right)} $$ ,
We have,
$\int {{{\sec }^2}x.dx} = \tan x + c$ ,
Let us substitute the above formula to the equation.
We get,
$$ \Rightarrow \int {f\left( x \right).dx} = \dfrac{1}{a}.\tan \left( {\dfrac{{ax}}{2}} \right) + c$$
Therefore the correct option is 2.
So, the correct answer is “Option 2”.

Note: This is not a tough problem, we just need the right idea at the right time. If we did not get that idea it is not easy to solve the problem. The whole solution depends on the only idea of converting $1 + \cos ax$ . We must practice a lot so that we can get the idea quickly.