Question

How do you find the integral of the given function: $\dfrac{1}{\cos x}$?

Answer 1

Hint: We start solving the making use of the fact that $\dfrac{1}{\cos x}=\sec x$. We then multiply and divide the integrand with $\left( \sec x+\tan x \right)$ to proceed through the problem. We then assume $t=\sec x+\tan x$ and find $dt$ in terms of $dx$. We then these results in the integral and then make use of the fact that $\int{\dfrac{dx}{x}}=\ln x+C$ and then make the necessary calculations to get the required answer.

Complete step by step answer:
According to the problem, we are asked to find the integral of the given function $\dfrac{1}{\cos x}$.
Let us assume $I=\int{\dfrac{1}{\cos x}dx}$ ---(1).
We know that $\dfrac{1}{\cos x}=\sec x$. Let us use this result in equation (1).
$\Rightarrow I=\int{\sec xdx}$ ---(2).
Let us multiply and divide the integrand with $\left( \sec x+\tan x \right)$ in equation (2).
$\Rightarrow I=\int{\dfrac{\sec x\left( \sec x+\tan x \right)}{\left( \sec x+\tan x \right)}dx}$.
$\Rightarrow I=\int{\dfrac{\left( {{\sec }^{2}}x+\sec x\tan x \right)}{\left( \sec x+\tan x \right)}dx}$ ---(3).
Let us assume $t=\sec x+\tan x$ ---(4). Now, let us apply a differential on both sides of equation (4).
$\Rightarrow dt=d\left( \sec x+\tan x \right)$ ---(5).
We know that $d\left( a+b \right)=da+db$. Let us use this result in equation (4).
$\Rightarrow dt=d\left( \sec x \right)+d\left( \tan x \right)$ ---(6).
We know that $d\left( \sec x \right)=\sec x\tan xdx$ and $d\left( \tan x \right)={{\sec }^{2}}xdx$. Let us use these results in equation (6).
$\Rightarrow dt=\sec x\tan xdx+{{\sec }^{2}}xdx$.
$\Rightarrow dt=\left( \sec x\tan x+{{\sec }^{2}}x \right)dx$ ---(7).
Let us substitute equations (4) and (7) in equation (3).
$\Rightarrow I=\int{\dfrac{dt}{t}}$ ---(8).
We know that $\int{\dfrac{dx}{x}}=\ln x+C$. Let us use this result in equation (8).
$\Rightarrow I=\ln t+C$.
From equation (3), we have $t=\sec x+\tan x$.
$\Rightarrow I=\ln \left( \sec x+\tan x \right)+C$.

$\therefore $ We have found the integral of the given function $\dfrac{1}{\cos x}$ as $\ln \left( \sec x+\tan x \right)+C$.

Note: We should perform each step carefully in order to avoid confusion and calculation mistakes. We should not forget to add constant of integration while solving the problems related to indefinite integrals. We should not forget to replace $t$ with $\sec x+\tan x$ after equation (8), as this is the common mistake done by students. Similarly, we can expect problems to find the integral of the function $\dfrac{1}{\tan x}$.