Question

Use the Exponential Rule to find the indefinite integral. $$ \int e^{-0.25 x} d x $$

Short answer

The indefinite integral of \( e^{-0.25x} \) is \( -4 e^{-0.25x} + C \).

Step-by-step solution

The Exponential Rule for Integration

The exponential rule for integration states that the integral of \( e^x \) with respect to \( x \) is \( e^x \). So, the integral of \( e^{ax} \) with respect to \( x \) is \( \frac{1}{a}e^{ax} + C \), where \( C \) is the constant of integration. Apply this rule to \( \int e^{-0.25 x} d x \).

Apply the Rule

Applying this rule to this function, where \( a = -0.25 \), we get \( \frac{1}{-0.25}e^{-0.25 x} + C \). Simplify this further.

Simplify

The expression simplifies to \( -4 e^{-0.25 x} + C \).

Key concepts

Exponential Rule for Integration

Integration is a fundamental technique in calculus that is used to find the antiderivative or the area under a curve. When it comes to exponential functions, the Exponential Rule for integration becomes particularly valuable. This rule is applied to functions of the form e^{ax}, where a is a constant.

To integrate an exponential function, you can use the formula:
\[\int e^{ax} dx = \frac{1}{a}e^{ax} + C,\]
where C represents the constant of integration. This formula is derived from the fact that the derivative of e^{ax} is ae^{ax}, hence reversing this process gives us the integral.

• Identify the constant a in the exponent of e.
• Write down the integral according to the formula, including the factor of 1/a.

In our exercise, we have a = -0.25, making the antiderivative -4e^{-0.25x}. This simplification process is critical to mastering the integration of exponential functions.

Integration Techniques

There is an array of techniques available for solving integrals in calculus. Applied adeptly, these techniques can simplify complex integrals into basic forms that are easier to evaluate.

• Substitution: Also known as the 'reverse chain rule', it involves substituting part of the integral with a new variable to simplify the integral.
• Integration by Parts: Based on the product rule for differentiation, this technique is useful for integrating products of functions.
• Partial Fractions: This technique breaks down a complex fraction into simpler partial fractions, making integration possible.
• Trigonometric Substitution: Substitution using trigonometric identities to simplify integrals involving square roots.
• Numerical Integration: When a function is too difficult to integrate analytically, numerical methods can approximate the area under the curve.

Each technique has specific scenarios where it is most effective, and selecting the right method is key to arriving at a correct and simplified result.

In the context of our exponential function, the Exponential Rule is straightforward and efficient, hence we do not require additional techniques for this problem.

Constant of Integration

When we find the indefinite integral of a function, we are essentially looking for all the possible antiderivatives. Since the process of differentiation washes away constant terms (as their derivative is zero), when we integrate, we must account for these lost constants. We do this by adding the constant of integration, denoted by C.

Here are a few points to keep in mind about the constant of integration:

• It represents an infinite number of possible values, reflecting the fact that there are infinitely many antiderivatives.
• In the absence of initial conditions or boundary conditions, the constant remains undetermined.
• When evaluating definite integrals, the constant of integration cancels out, and as such, it is not included.

In our exercise, after applying the Exponential Rule for integration, we added the constant of integration to the result, yielding -4e^{-0.25x} + C. This inclusion ensures that we have accounted for all antiderivatives of the exponential function in our solution.