Question

(a) integrate to find \(F\) as a function of \(x\) and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). $$ F(x)=\int_{\pi / 3}^{x} \sec t \tan t d t $$

Short answer

The antiderivative of the given function is \(F(x) = \sec x - \sec(\pi / 3)\), and its derivative, which confirms the Second Fundamental Theorem of Calculus, is \(F'(x) = \tan x \sec x\).

Step-by-step solution

Integration

Firstly, integrate the function \(F(x) = \int_{\pi / 3}^{x} \sec t \tan t dt \). Recall that integration of \( \sec x \tan x\) is \( \sec x \). Therefore, \( F(x) = [ \sec t]_{\pi/3}^{x}\). Hence, after substituting the limits, obtain \(F(x) = \sec x - \sec (\pi / 3)\).

Differentiation

In the second part, demonstrate the Second Fundamental Theorem of Calculus by differentiating the result from part (a), which is \(F(x) = \sec x - \sec(\pi / 3)\). The derivative of \( \sec x \) is \( \tan x \sec x \), and the derivative of a constant is 0. Therefore, obtain \( F'(x) = \tan x \sec x\).

Verification

To confirm that everything was done correctly, compare the derivative of the function found in step 2, \(F'(x) = \tan x \sec x \), with the original integrand \( \sec t \tan t\). The integration and differentiation operations are the inverse of each other. Thus, when differentiating the integrated function, should get back the original function. Indeed, \(F'(x) = \sec x \tan x\), which matches the original function.

Key concepts

Integration

Integration is a fundamental concept in calculus that involves finding the antiderivative or the 'area under the curve' of a function. It is often used to determine the total accumulation of quantity, such as distance traveled over time, from a rate of change (like velocity). In our exercise, we are given the integrand \( \sec t \tan t \). This specific integrand is a notable one, as it represents the derivative of the secant function, \( \sec t \). When finding \( F(x) = \int_{\pi / 3}^{x} \sec t \tan t \, dt \), we integrate the function with respect to \( t \) and evaluate it from \( \pi/3 \) to \( x \), effectively calculating the area under the curve between these two values.

To integrate \( \sec t \tan t \), we recognize it as the derivative of \( \sec t \) and, therefore, its antiderivative is \( \sec t \) itself. The process involves using antiderivative rules and knowing specific integrals by memory or through the use of integral tables. After applying the limits of integration, we get the function \( F(x) = \sec x - \sec(\pi / 3) \), which represents the accumulated area. Integration allows us to move from a rate of change (represented by the integrand) to the total quantity (represented by our function \( F(x) \) after integration).

Differentiation

Differentiation is another core concept in calculus and it denotes the process of finding the rate of change or the derivative of a function. If integration can be viewed as 'summing up' over an interval, differentiation is about finding the instantaneous change at a single point. It’s like zooming in on a curve until it starts to resemble a straight line, and then measuring the slope of that line.

In part (b) of our exercise, we differentiated the result from part (a), the function \( F(x) = \sec x - \sec(\pi / 3) \). Differentiating \( \sec x \) uses one of the essential differentiation formulas, yielding \( \tan x \sec x \). The constant term \( \sec(\pi / 3) \) disappears upon differentiation because the slope of a constant is always zero (as it represents a horizontal line with no change). The derivative of \( F(x) \) is, therefore, \( F'(x) = \tan x \sec x \), which reflects the rate of change of the function \( F(x) \) with respect to \( x \) and should match the original integrand according to the Second Fundamental Theorem of Calculus.

The Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus bridges the concepts of integration and differentiation, showing they are inverse processes. It states that if you have a continuous function \( f \) defined on an interval \( [a, b] \) and if \( F \) is the antiderivative of \( f \), then \( \int_{a}^{b} f(t) \, dt = F(b) - F(a) \). Moreover, the theorem also tells us that the derivative of \( F \) with respect to \( x \) is the original function \( f(x) \).

Applying this to our problem, we notice that \( F(x) = \int_{\pi / 3}^{x} \sec t \tan t \, dt \) and its derivative \( F'(x) \) gives us back the original integrand, \( \sec x \tan x \) - validating the process. This result becomes a powerful tool as it allows us to evaluate definite integrals using antiderivatives. The Second Fundamental Theorem of Calculus also provides the foundation for many techniques and applications of integration, including the calculation of areas, volumes, and other physical quantities from a known rate of change.