Calculate the integral: integral from - pi/4 to (pi/4)

Calculate the integral: $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{dx}{\cos^2 x}$.

We will use the Newton-Leibniz formula after recognizing the antiderivative.

Step 1: Recognize the integrand

The integrand can be written as:

\[\frac{1}{\cos^2 x} = \sec^2 x\]

Step 2: Find the antiderivative

We know that:

\[\frac{d}{dx}[\tan x] = \sec^2 x = \frac{1}{\cos^2 x}\]

Therefore, the antiderivative is:

\[F(x) = \tan x + C\]

Step 3: Apply the Newton-Leibniz formula

\[\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{dx}{\cos^2 x} = \left[\tan x\right]_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\]

Step 4: Evaluate at the upper limit $x = \frac{\pi}{4}$

\[\tan\left(\frac{\pi}{4}\right) = 1\]

Step 5: Evaluate at the lower limit $x = -\frac{\pi}{4}$

\[\tan\left(-\frac{\pi}{4}\right) = -\tan\left(\frac{\pi}{4}\right) = -1\]

(using the fact that tangent is an odd function: $\tan(-x) = -\tan(x)$)

Step 6: Calculate the definite integral

\[\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{dx}{\cos^2 x} = \tan\left(\frac{\pi}{4}\right) - \tan\left(-\frac{\pi}{4}\right) = 1 - (-1) = 2\]

Alternative observation using symmetry:

We can also note that $f(x) = \frac{1}{\cos^2 x} = \sec^2 x$ is an even function because $f(-x)=\frac1{\cos^2(-x)}=\frac1{\cos^2x}=f(x)$ (since cosine is even). Therefore:

\[\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{dx}{\cos^2 x} = 2\int_0^{\frac{\pi}{4}} \frac{dx}{\cos^2 x} = 2\left[\tan x\right]_0^{\frac{\pi}{4}} = 2(\tan\left(\frac{\pi}{4}\right) - \tan(0)) = 2(1-0) = 2\]

Geometric interpretation:

The function $y = \sec^2 x = \frac{1}{\cos^2 x}$ is always positive on $\left(-\frac\pi2,\frac\pi2\right)$, has a minimum value of 1 at $x=0$, increases as we move away from $x=0$ in either direction, and is symmetric about the $y$-axis (even function). At $x=\pm\frac\pi4$: $\sec^2\left(\frac\pi4\right)=\frac1{(\sqrt2/2)^2}=2$.

The integral represents the area under this curve from $x = -\frac{\pi}{4}$ to $x = \frac{\pi}{4}$, which equals 2.

Final answer:

\[\boxed{\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{dx}{\cos^2 x} = 2}\]