Problem 71 - Applied Calculus

Question

Find the indefinite integrals. $$\int 100 e^{4 x} d x$$

Step-by-Step Solution

Answer

The indefinite integral is \(25 e^{4x} + C\).

Step 1: Identify the Integral Formula

The given problem requires you to find the indefinite integral of the function \(100 e^{4x}\). The indefinite integral of \(e^{ax}\) is \(\frac{1}{a}e^{ax} + C\), where \(C\) is the constant of integration and \(a\) is a constant.

Step 2: Apply the Integral Formula

Here \(a = 4\), so the integration of \(e^{4x}\) is \(\frac{1}{4}e^{4x} + C\). Because of the constant multiple rule of integration, you can factor out the constant 100.

Step 3: Calculate the Integral

By bringing out the constant 100 from the integral, we have \(100 \int e^{4x} \, dx\). Now, integrate \(e^{4x}\) to get \(\frac{1}{4}e^{4x} + C\). Thus, the integral becomes \(100 \times \frac{1}{4}e^{4x} + 100C\).

Step 4: Simplify the Expression

Simplify the expression obtained: \(\frac{100}{4}e^{4x} + C'\), where \(C' = 100C\). This gives \(25 e^{4x} + C\). Thus, the indefinite integral is \(25 e^{4x} + C\).

Key Concepts

Integration Techniques

When tackling indefinite integrals, you'll often need to apply different integration techniques to solve problems. In calculus, integration techniques are diverse, from basic rules to more advanced methods. One fundamental rule is the **constant multiple rule**, which allows you to factor constants out of an integral. It simplifies the process and makes calculations more manageable. For example, in this exercise, you have the integral \(\int 100 e^{4x} \, dx\). You can factor the 100 out, leaving you with 100 multiplied by the integral of \(e^{4x}\).

Another essential technique is recognizing patterns that match known integral formulas. The exponential function \(e^{ax}\) has a straightforward integral formula, \(\frac{1}{a}e^{ax} + C\), where \(a\) is a constant.

By combining these techniques, you're able to break the problem into simpler parts, perform calculations, and then piece everything back together. Remember to always account for the constant of integration, which we'll discuss further in a separate section.

Exponential Functions

Exponential functions play a crucial role in calculus. They often appear in real-world applications, making them essential to understand. The most common exponential function is \(e^x\), where \(e\) is a mathematical constant approximately equal to 2.71828. This function grows rapidly and has unique properties.

### Properties of Exponential Functions

• The derivative of \(e^{ax}\) is \(ae^{ax}\).
• The integral of \(e^{ax}\) is \(\frac{1}{a}e^{ax} + C\).

Understanding these properties is key when integrating exponential functions.

In our exercise, we had \(100 e^{4x}\). Recognizing that it's in the form \(e^{ax}\) with \(a=4\) allows us to use the formula for integration directly. The process involves substituting \(a=4\) into the integral formula, giving \(\frac{1}{4}e^{4x} + C\), and then multiplying the entire expression by 100, due to the constant multiple rule.

Constant of Integration

In indefinite integrals, the constant of integration \(C\) is a vital component. It accounts for any constant value that disappears when differentiating a function. Since indefinite integrals are essentially "undoing" differentiation, the result must include \(C\) to acknowledge all possible original functions.

### Importance of the Constant of Integration

• Represents an infinite set of possible antiderivatives.
• Ensures the integral is complete and accurate.
• Indispensable in solving differential equations and initial value problems.

In our solution, the constant \(C' = 100C\) arises after factoring out the 100 during integration. It ensures that our solution captures every potential antiderivative of the function. When solving indefinite integrals, always include \(C\) in your final answer to reflect this mathematical truth.