Problem 38 - Calculus for Biology and Medicine

Question

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=2 e^{-3 x}+\sec ^{2}\left(-\frac{x}{2}\right) $$

Step-by-Step Solution

Answer

The general antiderivative is \(-\frac{2}{3} e^{-3x} + 2\tan\left(\frac{x}{2}\right) + C\).

Step 1: Identify Each Function Component

The given function is composed of two parts: \(2 e^{-3x}\) and \(\sec^2\left(-\frac{x}{2}\right)\). The task is to find the general antiderivative of the entire expression.

Step 2: Antiderivative of the First Term

For the first term, \(2 e^{-3x}\), the antiderivative is found by recognizing it as a basic exponential function. The antiderivative of \(e^{ax}\) is \(\frac{1}{a} e^{ax}\). So, for \(2 e^{-3x}\), the antiderivative is \(-\frac{2}{3} e^{-3x} + C_1\), where \(C_1\) is a constant of integration.

Step 3: Antiderivative of the Second Term

For the second term, \(\sec^2\left(-\frac{x}{2}\right)\), recall that the derivative of \(\tan(x)\) is \(\sec^2(x)\). Therefore, the antiderivative is \(-2\tan\left(-\frac{x}{2}\right) + C_2\), where \(C_2\) is another constant of integration. The factor \(-2\) arises from the chain rule because the argument of \(\sec^2\) is multiplied by \(-\frac{1}{2}\).

Step 4: Combine Antiderivatives

Combine the antiderivatives from the two terms to express the general antiderivative of the entire function: \(-\frac{2}{3} e^{-3x} - 2\tan\left(-\frac{x}{2}\right) + C\), where \(C\) is a constant representing the sum of \(C_1 + C_2\).

Step 5: Simplification

Recognize that \(\tan(-u) = -\tan(u)\), enabling you to further simplify \(-2\tan\left(-\frac{x}{2}\right)\) to \(2\tan\left(\frac{x}{2}\right)\). Thus, the antiderivative simplifies to \(-\frac{2}{3} e^{-3x} + 2\tan\left(\frac{x}{2}\right) + C\).

Key Concepts

Antiderivative

The antiderivative is also known as the indefinite integral. It's a central concept in calculus, used to find a function whose derivative is the given function.
When we have a function like \( f(x) = 2 e^{-3x} + \sec^2\left(-\frac{x}{2}\right) \), we aim to find a new function \( F(x) \) such that \( F'(x) = f(x) \). Finding an antiderivative requires familiarity with the basic rules of integration and the derivatives of standard functions.

• For example, the antiderivative of \( e^{ax} \) is \( \frac{1}{a} e^{ax} \), where \( a \) is a constant.
• For trigonometric functions, like \( \sec^2(x) \), knowing their derivatives helps; \( \sec^2(x) \) is the derivative of \( \tan(x) \).

The final expression for the antiderivative will include a constant \( C \), which represents any constant term that could be added without affecting the derivative. In our example, the combined antiderivative is \(-\frac{2}{3} e^{-3x} + 2\tan\left(\frac{x}{2}\right) + C\).
This process highlights how integration serves as a tool to reconstruct the original function from its rate of change.

Exponential Functions

Exponential functions are powerful mathematical tools that describe exponential growth or decay. The general form of an exponential function is \( f(x) = e^{bx} \), where \( e \) is the base of the natural logarithms, approximately equal to 2.718.
These functions have unique properties:

• The derivative of \( e^{bx} \) is \( be^{bx} \).
• This implies that the function grows or decays at a rate proportional to its current value.

When considering antiderivatives, we use the reverse of the derivative rule.
Thus, for a function like \( 2e^{-3x} \), you find its antiderivative by making it \( -\frac{2}{3}e^{-3x} \). This accounts for the coefficient \( -3 \) in the exponent by adjusting with a factor of \( \frac{1}{a} \).
Exponential functions are pervasive in natural sciences and financial modeling because they naturally model phenomena such as population growth, radioactive decay, and interest compounding.

Trigonometric Functions

Trigonometric functions are related to angles and the geometry of circles. Examples include sine, cosine, and tangent.
The function \( \sec^2(x) \), for example, is linked to the tangent function. Specifically, the derivative of \( \tan(x) \) is \( \sec^2(x) \). This makes finding its antiderivative straightforward:

• \( \sec^2\left(-\frac{x}{2}\right) \) becomes \(-2\tan\left(-\frac{x}{2}\right)\).
• Applying the identity \( \tan(-u) = -\tan(u) \) simplifies this further to \( 2\tan\left(\frac{x}{2}\right) \).

This example demonstrates the utility of knowing derivative and integration rules for trigonometric functions.
They apply widely in fields such as engineering, physics, and computer graphics due to their cyclical nature, which models periodic processes effectively. Understanding these principles allows you to analyze wave patterns, vibrations, and even signal processing.