Question

How do you find the integral of $\dfrac{1}{{{{(\cos x)}^2}}}$?

Answer 1

Hint:
As we know about the integral, it is used for finding the areas, volumes and for central points etc. We use it for finding the integration of a particular given identity. It is denoted by the $\int {} $ sign; it is done with respect to a variable which can be $x,y,z$ etc.

Complete step by step solution:
Given that –
According to question integrate ${(\dfrac{1}{{\cos x}})^2}$
Let – $I = {(\dfrac{1}{{\cos x}})^2}$
Now we can write it as ${(\dfrac{1}{{\cos x}})^2} = \dfrac{1}{{{{\cos }^2}x}}$ because both are same in the trigonometry identity
We know that the three basic identity of trigonometry are First is ${\sin ^2}x + {\cos ^2}x = 1$ and second is ${\tan ^2}x + 1 = {\sec ^2}x$ and the third identity is the ${\cot ^2}x + 1 = \cos e{c^2}x$.
We have to know the conversion formula in the trigonometry which are very useful so now we will see them $\dfrac{1}{{\sin x}} = \cos ecx$ and $\dfrac{1}{{\cos x}} = secx$ and $\dfrac{1}{{\tan x}} = \cot x$ now we will use our second formula which is $\dfrac{1}{{\cos x}} = secx$
Now we will integrate the $I$ with respect to $x$
$ = \int I $
$ = \int {\dfrac{1}{{{{\cos }^2}x}}dx} $
Now we will put the value of $\dfrac{1}{{\cos x}} = secx$ in the above equation then we get
$ = \int {({{\sec }^2}x)} dx$
Now we know that we can distribute integration on addition and subtraction so we will integrate it because $\int {({{\sec }^2}x)} dx = \tan x + c$ then
$ = \int {({{\sec }^2}x)} dx$
Now we know that the integration of $\int {{{\sec }^2}} xdx = \tan x$ now we will put these values in above equation then we get
$ = \tan x + c$
Where $c$is the constant value which we get when we do integration of any identity
Therefore the integration of ${(\dfrac{1}{{\cos x}})^2}$ is the $\tan x + c$ which is our required answer.

Note:
We can solve it directly by only using the second conversion formula by just putting the value of $\dfrac{1}{{\cos x}} = secx$ and then we get the same answer in the three or four step, if this question is in the objective type otherwise follow the above method.