Question

Integrate $\dfrac{1+\tan x}{1-\tan x}dx$

Answer 1

In order to integrate $\dfrac{1+\tan x}{1-\tan x}dx$, firstly we will be expressing $\tan x$ in terms of $\sin x$ and $\cos x$ in both numerator and denominator and then we will be solving it accordingly. After solving we will consider $\cos x-\sin x=t$ and then upon substituting and integrating this function, we will be obtaining our required result.

Complete step-by-step solution:
Now let us have a brief regarding integration. It is nothing but calculating the integral. The integrals are usually termed regarding the definite integrals and indefinite integrals are used for antiderivatives. Integration is the reverse process of differentiation. There are two types of integrals. They are: definite and indefinite integrals. Integration can be performed in different methods. They are: Integration by Substitution, Integration by Parts, Integration using Trigonometric Identities, Integration of Some Particular Function and Integration by Partial Fraction.
Now let us start integrating $\dfrac{1+\tan x}{1-\tan x}dx$
We have, $I=\dfrac{1+\tan x}{1-\tan x}dx$
Firstly, let us express $\tan x$ in terms of $\sin x$ and $\cos x$. We get
$\Rightarrow \int{\dfrac{1+\tan x}{1-\tan x}}dx=\int{\dfrac{1+\dfrac{\sin x}{\cos x}}{1-\dfrac{\sin x}{\cos x}}}dx$
Upon solving this, we get
$\Rightarrow \int{\dfrac{\cos x+\sin x}{\cos x-\sin x}dx}$
Now let us consider $\cos x-\sin x=t$
Upon differentiating this, we obtain

& \Rightarrow \cos x-\sin x=t \\\ & \Rightarrow -\sin x-\cos x=dt \\\ & \Rightarrow \sin x+\cos x=-dt \\\ \end{aligned}$$ Now let substitute and the integrate this, $$I=-\int{\dfrac{dt}{t}}$$ $$\Rightarrow -\log t+c$$ We know that $$\cos x-\sin x=t$$, so we get $$\Rightarrow -\log \left| \cos x-\sin x \right|+c$$ **Note:** While performing the integration, we must be aware of the sum rule, power rule, difference rule, etc. We must think of the method to be followed for integrating as we have followed the method of substitution in the above problem. While solving, we must cross check with the integration rules for easy solving of the functions given. We can apply integration in finding the areas, volumes, central points and many more.