Evaluate the integral: integral tan² x dx
Evaluate the integral: $$ \int \tan^2 x \, dx. $$
Explanation of Method
The Pythagorean identity $\tan^2 x = \sec^2 x - 1$ converts a power of tangent, which has no elementary antiderivative on its own, into a difference of two elementary integrals: the standard derivative-of-tangent form $\sec^2 x$ and the constant function $1$.
- Rewrite $\tan^2 x$ using the Pythagorean identity.
- Split the integral into two simpler integrals.
- Integrate each term using standard results.
- Combine the results and add the constant of integration.
Step 1 · Use the trigonometric identity for $\tan^2 x$
The trigonometric identity for $\tan^2 x$ is:
$$ \tan^2 x = \sec^2 x - 1. $$
Using this identity, the integral becomes:
$$ \int \tan^2 x\,dx = \int (\sec^2 x - 1)\,dx. $$
Step 2 · Split the integral
The integral is now split into two simpler integrals:
$$ \int \tan^2 x\,dx = \int \sec^2 x\,dx - \int 1\,dx. $$
First term: $\displaystyle \int \sec^2 x\,dx$ is a standard result:
$$ \int \sec^2 x\,dx = \tan x. $$
Second term: $\displaystyle \int 1\,dx$:
$$ \int 1\,dx = x. $$
Step 3 · Combine the results
Combining the results of the two integrals:
$$ \int \tan^2 x\,dx = \tan x - x + C, $$
where $C$ is the constant of integration.
Summary Table
| Term | Antiderivative |
|---|---|
| $\sec^2 x$ | $\tan x$ |
| $-1$ | $-x$ |
Additional Notes
- The domain of $I$ excludes $x = \dfrac{\pi}{2} + k\pi$ for integer $k$, where $\tan x$ and $\sec^2 x$ are undefined.
- This is a standard reduction: any power of $\tan x$ or $\cot x$ can be lowered using $\tan^2 x=\sec^2 x-1$ or $\cot^2 x=\csc^2 x-1$ repeatedly.
- The result can be checked by differentiating $\tan x - x$ and confirming it returns $\sec^2 x - 1 = \tan^2 x$.
Result
The solution to the integral is: