Problem 10 - Calculus: One and Several Variables

Question

Calculate. $$\int_{-\pi / 4}^{\pi / 4} \frac{d x}{\cos ^{2} x}$$

Step-by-Step Solution

Answer

The definite integral of the given function over the interval \(\left(-\frac{\pi}{4}, \frac{\pi}{4}\right)\) is equal to 2.

Step 1: Identify the antiderivative.

The antiderivative of \(\frac{1}{\cos^2{x}}\) is given by the function \(\tan{x}\). This is because the derivative of \(\tan{x}\) is \(\sec^2{x}\), which is equal to \(\frac{1}{\cos^2{x}}\).

Step 2: Evaluate the antiderivative at the interval endpoints.

Now we need to evaluate the antiderivative function \(\tan{x}\) at the endpoints of the given interval, which are \(-\frac{\pi}{4}\) and \(\frac{\pi}{4}\). This will give us: \(\tan{\left(-\frac{\pi}{4}\right)}=-1\) \(\tan{\left(\frac{\pi}{4}\right)}=1\)

Step 3: Calculate the definite integral.

Finally, we will find the difference between the values of the antiderivative function at the interval endpoints: \[\int_{-\pi / 4}^{\pi / 4} \frac{d x}{\cos ^{2} x} = \tan{\left(\frac{\pi}{4}\right)} - \tan{\left(-\frac{\pi}{4}\right)} = 1 - (-1) = 2\] The definite integral of the given function over the interval \(\left(-\frac{\pi}{4}, \frac{\pi}{4}\right)\) is equal to 2.

Key Concepts

Integral Calculus

Integral calculus is a crucial part of calculus that involves the concept of integration. While differentiation deals with finding the rate of change, integration is about finding the total accumulation of quantities. In simple terms, integration allows us to calculate areas under curves, among other things. It involves concepts like the antiderivative and definite integral.

• An antiderivative of a function is another function whose derivative matches the original function.
• A definite integral is an integral with upper and lower limits, in this case, from \(-\frac{\pi}{4}\) to \(\frac{\pi}{4}\).

By finding the antiderivative of a function and then evaluating it at these limits, we can compute the accumulation, or total change, of the original function over that interval.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are fundamental in mathematics, particularly in calculus. In this exercise, we deal with the cosine function, specifically \(\cos^2 x\).

• The cosine function describes the ratio of the adjacent side to the hypotenuse in a right triangle.
• The function \(\frac{1}{\cos^2 x}\) is actually equivalent to \(\sec^2 x\), where secant (sec) is the reciprocal of cosine.

Trigonometric functions follow specific patterns, useful in calculus, such as periodicity and symmetry. In the context of this problem, knowing that the antiderivative of \(\sec^2 x\) is \(\tan x\) helps immensely in solving the integral.

Antiderivative Calculation

To solve integrals efficiently, understanding how to find antiderivatives is key. The antiderivative is essentially the reverse process of differentiation. The goal is to find a function whose derivative matches the given function.

• In our exercise, the function \(\frac{1}{\cos^2 x}\) is equivalent to \(\sec^2 x\).
• We recognize that the antiderivative of \(\sec^2 x\) is \(\tan x\), because the derivative of \(\tan x\) is \(\sec^2 x\).

Once the antiderivative is found, it is evaluated at the given limits of integration, which in this problem are \(\left(-\frac{\pi}{4}, \frac{\pi}{4}\right)\). By applying these concepts, you can compute the overall change or area for complicated functions by reducing them to simpler expressions.