Question

How do you integrate $\tan (x)$?

Answer 1

Hint: In the given question, we have been given an algebraic expression that is to be integrated. And can be solved by opening $\tan (x)$ in terms of $\sin (x)$ and $\cos (x)$ and then trying to integrate the problem.

Complete step by step solution:
The given expression is $\tan (x)$ .
As we know, \[tan(x)=\dfrac{\sin (x)}{\cos (x)}\]
Hence, now we have to integrate,
\[\int \dfrac{\sin (x)}{\cos (x)}dx\]
Let \[u=cos(x)\]
On differentiating both sides, we get
\[\Rightarrow du=-sin(x)dx\]
Now, on substituting the values on above equation we will get,
\[\therefore -\int \dfrac{du}{u}=-ln(u)+C\]

Hence, \[\int \tan x dx=-ln(cosx)+C\].

Additional Information:
Integration is the opposite of differentiation. In differentiation, we “break” things for examining how they behave separately. While, in integration, we combine the expressions so as to see their collective behavior.

Note:
In these types of questions, an algebraic expression can be given which was to be integrated. To do that, we have to replace the square root bracket by expressing it in the form of power and then apply the integration formula to solve it. It is thus, really important that we know the formula of the integration, how to transform one quantity (algebra) to another (trigonometric), and how to use it in this case. Care must be taken when we are performing the steps as that is the point where things get tricky and there are chances of getting it wrong.