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Define $f:\mathbb R \rightarrow \mathbb R$ by $f(x)=\frac{1}{1+3\cos^{2}x}$.

Since $f$ is continuous, $f$ has an antiderivative. That is, there exists some function $F:\mathbb R \rightarrow \mathbb R$ such that $F^{\prime}=f$.

Using the substitution $u=\tan x$ and doing some work, we find that a possible antiderivative of $f$ is the function $g$ defined by $g(x)=\frac{1}{2}\tan^{-1}\frac{\tan x}{2}$.

However, the problem with $g$ is that it's not defined at odd multiples of $\pi/2$, while the $F$ we're seeking is defined on $\mathbb R$.

So, what is $F$ and how do we find it?

(Or is there a mistake somewhere in my argument above?)

  • I won't mark this one as a duplicate since the question statements are rather different, but the issue is exactly the same as the one in this question, which has good* answers taking various approaches, involving a similar integral: https://math.stackexchange.com/questions/3275770/how-to-integrate-int-02-pi-frac1-sin4x-cos4-x-dx/3275818 (*One of the answers is mine, I make no claims about its goodness.) – Travis Willse Feb 23 '20 at 04:21

1 Answers1

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For each odd integer $k$, define $f_k:((k-2)\pi/2,k\pi/2) \rightarrow \mathbb R$ by $f_k(x)=\frac{1}{1+3\cos^{2}x}$.

As you've found, an antiderivative of $f_k$ is $g_k:((k-2)\pi/2,k\pi/2) \rightarrow \mathbb R$ defined by $g_k(x)=\frac{1}{2}\tan^{-1}\frac{\tan x}{2}$.

And as you've also correctly stated, we know by the FTC that $f$ must have an antiderivative $F$.

So, we guess that such a function $F$ can be constructed by "moving up or down" the functions $g_k$ and "stitching" them together, and also "filling in the holes" at each odd multiple of $\pi/2$.

By examining the graphs of each $g_k$, we come up with this guess:

For each odd integer $k$, define $F$ by $$F(x)=\begin{cases} \left(k-1\right)\frac{\pi}{2}, & \text{ for }x=\left(k-2\right)\frac{\pi}{2},\\ \frac{1}{2}\tan^{-1}\frac{\tan x}{2}+k\frac{\pi}{4}, & \text{ for }x\in\left(\left(k-2\right)\frac{\pi}{2},k\frac{\pi}{2}\right)\\ k\frac{\pi}{2}, & \text{ for }x=k\frac{\pi}{2}. \end{cases},$$

To show that the above guess is correct, we can verify that $F^{\prime}=f$ (this will involve more work).

Similar questions: 1, 2, 3. See also Jeffrey and Rich (1994, "The Evaluation of Trigonometric Integrals Avoiding Spurious Discontinuities") and Jeffrey (1994, "The Importance of Being Continuous").