Problem 10 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Compute the integral. \(\int \frac{\sec ^{2} t}{5} d t\)

Step-by-Step Solution

Answer

The integral of \(\int \frac{\sec^2t}{5} dt\) is \(\frac{1}{5} tan(t) + C\).

Step 1: Consider the integral of sec^2(x)

The basic integral of \(\sec^2(x)\) in the standard integral table is \(tan(x)\) + C where C is the constant of integration. This result is derived from the fact that the derivative of \(tan(x)\) with respect to x is \(\sec^2(x)\).

Step 2: Compute the given integral

Now, consider the given integral which is \(\int \frac{\sec^2t}{5} dt\). Based on the result from step 1, one can just slot \(tan(t)\) in the place of \(\sec^2(t)\) to integrate. Therefore, the integral becomes \(\frac{1}{5} \int tan(t) dt\).

Step 3: Simplify the integral

Now, multiplying the integral by \(\frac{1}{5}\) which is a constant we get \(\frac{1}{5} tan(t) + C\). Here, \(C\) represents a constant.

Key Concepts

Integral of sec^2(x)

Understanding the integral of the function sec²(x) is crucial for solving various calculus problems. The function sec(x) denotes the secant of x, which is a trigonometric function equivalent to the reciprocal of the cosine function. When raised to the power of 2, it represents the square of the secant function. The integration of sec²(x) is rather straightforward because it is one of the standard integrals that you will often encounter in calculus.

Mathematically, the integral of sec²(x) with respect to x is given by:
\[\int \sec^2(x) \, dx = \tan(x) + C\]
where C is the integration constant. This formula is a direct result of the fact that the derivative of the tangent function, \(\tan(x)\), with respect to x is sec²(x). Therefore, the antiderivative (or integral) of sec²(x) is just the tangent function plus a constant.

Antiderivative of tan(x)

The antiderivative of a function is another function that, when differentiated, yields the original function. In the case of the tangent function, tan(x), finding its antiderivative is connected to the integral of sec²(x). As we have seen in the previous section, sec²(x) is the derivative of tan(x).

This relationship is essential in calculus because it provides a direct method to evaluate the integrals involving the tangent function. For instance, the antiderivative of \(\tan(x)\) is \(\tan(x) + C\), where C denotes the integration constant. Whenever you integrate tan(x), you are finding a function that, when differentiated, would give you tan(x) again. It is the reason why the basic integral formula for sec²(x) is used for finding the antiderivative of tan(x).

Integration Constants

Integration constants are an important concept in calculus, especially when dealing with indefinite integrals. Indefinite integrals, unlike definite integrals, do not have set limits of integration and therefore yield a family of functions rather than a single value. This family of functions is represented by the integration constant, typically denoted as \(C\).

When you perform an integration, the constant C is added to represent all possible vertical shifts of the antiderivative. It is critical to always include the constant C when solving an indefinite integral, as it signifies the general solution to an antiderivative problem. Whether you are dealing with trigonometric integrals or any other type of function, the integration constant is a fundamental part of expressing the most general form of the solution.

Trigonometric Integrals

Trigonometric integrals involve the integration of trigonometric functions such as sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These types of integrals are commonplace in calculus and they have standard results that are widely used.

Understanding how to integrate trigonometric functions involves recognizing patterns and applying known formulas. For instance, you might encounter integrals involving powers of sine and cosine, products of different trigonometric functions, or even more complex trigonometric expressions.

An important strategy for evaluating trigonometric integrals is to use trigonometric identities to simplify the function into a form that is easier to integrate. For example, knowing that sec²(x) is the derivative of tan(x) simplifies the integration process for integrals involving sec²(x). As you study trigonometric integrals, you'll quickly learn these relationships and how to apply them effectively to solve integration problems.