Problem 70 - Applied Calculus

Question

Find the indefinite integrals. $$\int 25 e^{-0.04 q} d q$$

Step-by-Step Solution

Answer

The integral evaluates to \(-625 e^{-0.04q} + C\).

Step 1: Identify the Integral Type

The given integral is \( \int 25 e^{-0.04q} \, dq \). This is an exponential function, and the integral involves a constant multiplied by an exponential function of the form \( e^{ax} \).

Step 2: Apply the Exponential Integral Formula

The standard formula for integrating an exponential function \( e^{ax} \) is \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \), where \( C \) is the constant of integration.

Step 3: Identify Constants and Apply Formula

From the integral \( \int 25 e^{-0.04q} \, dq \), identify that \( a = -0.04 \) and the constant multiplier is 25. Apply the formula: \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \).

Step 4: Integrate the Function

Substitute \( a = -0.04 \) into the formula: \( \int 25 e^{-0.04q} \, dq = 25 \times \frac{1}{-0.04} e^{-0.04q} + C \).

Step 5: Simplify the Expression

Calculate \( \frac{1}{-0.04} \), which simplifies to \(-25\). Substitute to get: \( 25 \times (-25) e^{-0.04q} + C = -625 e^{-0.04q} + C \).

Key Concepts

Exponential Function

An exponential function is one of the fundamental building blocks of mathematics, especially in calculus. Exponential functions are expressed as \( e^{x} \) or \( e^{ax} \), where \( e \) is a mathematical constant approximately equal to 2.71828, and \( a \) can be any real number.

These functions have important properties:

• They grow or decay rapidly depending on whether \( a \) is positive or negative.
• They are continuous and smooth, with a unique feature of their derivative being proportional to themselves.

In integrals like \( \int 25 e^{-0.04 q} \, dq \), exponential growth or decay is adjusted by the constant multiplier in the exponent, in this case \(-0.04\). The negative sign here indicates that the function is exhibiting exponential decay.

Integration Formula

The integration formula for exponential functions \( \int e^{ax} \, dx \) is a key tool in calculus. When integrating \( e^{ax} \), the formula used is \[ \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \]
This formula shows that when you integrate an exponential function, the exponent does not change, but you divide by the constant \( a \) in the exponent.

• If \( a > 0 \), the function is growing, and if \( a < 0 \), it’s decaying.
• This integration can handle any real number multiplier, making it very versatile for different practical situations.

To apply this to the integral \( \int 25 e^{-0.04 q} \, dq \), you multiply the entire expression by the simplification of \( \frac{1}{a} \), which is \(-25\).

Constant of Integration

The constant of integration, \( C \), appears in the result of indefinite integrals. Because the process of integration is the inverse of differentiation, finding an indefinite integral implies determining the original function plus any constant.

Why is this needed?

• During differentiation, any constant term disappears. Therefore, when integrating, we have to account for any constant that might have been part of the original function.
• It reflects all possible vertical shifts of the antiderivative on a graph.

This means that for our integral, \( \int 25 e^{-0.04 q} \, dq = -625 e^{-0.04 q} + C \), the term \( C \) represents an infinite number of potential constants that could have been differentiated away in the process of arriving at \( 25 e^{-0.04 q} \). Always remember that the constant of integration ensures that all possible solutions to an indefinite integral are represented.