Problem 24 - Calculus: Early Transcendental Functions

Question

Find the general antiderivative. $$\int(2 \cos x-\sqrt{e^{2 x}}) d x$$

Step-by-Step Solution

Answer

The general antiderivative of the given function is \(2 \sin x - e^x + C\).

Step 1: Integral of Cosine Function

As you know, the antiderivative of the cosine function is the sine function. Therefore, the antiderivative of \(2 \cos x\) is simply \(2 \sin x\).

Step 2: Integral of Exponent Function

For the second term \(-\sqrt{e^{2x}}\), it is equivalent to \(-e^{x}\), as square root of \(e^{2x}\) is equal to \(e^x\). The antiderivative of \(e^x\) with respect to \(x\) is \(e^x\). Hence, the antiderivative of \(-e^x\) is \(-e^x\).

Step 3: Composition of the Antiderivatives

The antiderivative of the given function is the sum (or difference) of the two antiderivatives calculated in Steps 1 and 2. We therefore have the general antiderivative as: \(2 \sin x - e^x + C\), where \(C\) is the constant of integration.

Key Concepts

Integral Calculus

Integral calculus is a fundamental part of mathematics that deals with the accumulation of quantities and the areas under and between curves. When we talk about finding the antiderivative, we are essentially looking for a function whose derivative is the given function. This process is known as integration. In the exercise provided, students are tasked with integrating a combination of a cosine function and an exponential function.

When approaching integrals, it's important to identify and apply rules and techniques such as power rule, substitution, integration by parts, and specific formulas for trigonometric and exponential functions. The step-by-step solution demonstrates the application of these techniques, breaking down the problem into simpler parts to find the antiderivative of each term separately and combining them to arrive at the final solution.

Cosine Function

The cosine function is one of the primary trigonometric functions, often encountered in mathematics and physics. This function oscillates between -1 and 1, and it describes the relationship between the angles and lengths in a right triangle. For integration purposes, it's vital to remember that the antiderivative of the cosine function is the sine function.

Integration of Cosine

When dealing with the integral of the cosine function, such as the term 2 cos x in our exercise, we use the known result that the integral of cos x with respect to x is sin x. Hence, the integral of 2 cos x will simply be 2 sin x. This is a direct application of a basic integration rule which makes finding antiderivatives of trigonometric functions straightforward.

Exponential Function

Exponential functions, typically written as e^x where e is Euler's number (approximately 2.718), are famous for their unique property where the function's derivative is equal to itself. In the exercise, we see the square root of an exponential function, which complicates things slightly. However, with a solid understanding of exponents and their properties, this can be simplified.

Integral of Exponential Function

For -e^x, recognizing that the integral of e^x is itself e^x (since the derivative of e^x is e^x) allows us to integrate it directly, resulting in -e^x. In this case, the negative sign simply carries through the integration process. It's important for students to not only memorize this rule but also understand why it works to develop a deeper comprehension of exponential functions.