Integration Of Tanx 2: The Identity That Simplifies It
Integration of tan²x
The integral of tan²x is $$\tan x - x + C$$, and the quickest route is to use the identity $$\tan^2 x = \sec^2 x - 1$$. Because $$\int \sec^2 x\,dx = \tan x$$ and $$\int 1\,dx = x$$, the result follows directly.
Why this works
The key step is rewriting the trigonometric identity in a form that is easy to integrate. From $$1+\tan^2 x=\sec^2 x$$, we get $$\tan^2 x=\sec^2 x-1$$, which turns the problem into two standard integrals. This method is consistent across multiple calculus references.
Step-by-step method
- Start with $$\int \tan^2 x\,dx$$.
- Replace $$\tan^2 x$$ with $$\sec^2 x - 1$$.
- Split the integral: $$\int (\sec^2 x - 1)\,dx$$.
- Integrate each term to get $$\tan x - x + C$$.
Worked example
If you need a compact classroom-ready form, write the calculation as $$\int \tan^2 x\,dx = \int (\sec^2 x - 1)\,dx = \tan x - x + C$$. A quick derivative check confirms it: $$\frac{d}{dx}(\tan x - x)=\sec^2 x - 1=\tan^2 x$$.
Reference table
| Expression | Result | Reason |
|---|---|---|
| $$\int \tan x\,dx$$ | $$\ln|\sec x|+C$$ | Standard antiderivative. |
| $$\int \tan^2 x\,dx$$ | $$\tan x-x+C$$ | Use $$\tan^2 x=\sec^2 x-1$$. |
| $$\int \tan(2x)\,dx$$ | $$-\tfrac12\ln|\cos 2x|+C$$ | Different problem, requires substitution. |
Common mistake
Many students confuse $$\tan^2 x$$ with $$\tan(2x)$$, but they are different expressions and have different antiderivatives. The squared form integrates to $$\tan x-x+C$$, while the double-angle form gives a logarithmic answer.
In calculus, the fastest solution is often the one that turns a hard-looking expression into two familiar standard integrals.
Helpful tips and tricks for Integration Of Tanx 2 The Identity That Simplifies It
What is the integral of tan²x?
The integral of $$\tan^2 x$$ is $$\tan x - x + C$$.
What identity should I use?
Use $$1+\tan^2 x=\sec^2 x$$, then rewrite $$\tan^2 x$$ as $$\sec^2 x-1$$.
How can I check the answer?
Differentiate $$\tan x-x+C$$; the derivative simplifies to $$\sec^2 x-1$$, which equals $$\tan^2 x$$.
