The Integration of the Exponential Function e^(-x)

Learn the integration of the exponential function e⁻ˣ. This guide provides the integral ∫e⁻ˣ dx = -e⁻ˣ + C, essential for calculus students.

Integration of the Exponential Function $ e^{-x} $: Step-by-Step Guide

Step 1: Identify the Integral

Math: $\int e^{-x} \,dx$

Explanation: The integration of the exponential function $ e^{-x} $ begins by identifying the function we want to integrate. This is the starting point for understanding how to perform the integration of the exponential function $ e^{-x} $.

Step 2: Apply a Substitution

Math: $u = -x, du = -dx$

Explanation: In the integration of the exponential function $ e^{-x} $, a substitution is made to simplify the expression. This substitution is a key step in the integration of the exponential function $ e^{-x} $.

Step 3: Rewrite the Integral in Terms of $ u $

Math: $\int e^{u} \,du$

Explanation: The integral is rewritten in terms of $ u $, a crucial step in the integration of the exponential function $ e^{-x} $.

Step 4: Divide by -1

Math: $-\int e^{u} \,du$

Explanation: Since $ du = -dx $, we divide both sides of the integral by -1. This division is a vital algebraic step in the integration of the exponential function $ e^{-x} $.

Step 5: Integrate the Exponential Function

Math: $-e^{u} + C$

Explanation: The exponential function $ e^{u} $ is integrated with respect to $ u $, resulting in $-e^{u} + C$. This integration is a core part of the integration of the exponential function $ e^{-x} $.

Step 6: Substitute Back for $ x $

Math: $-e^{-x} + C$

Explanation: The final step in the integration of the exponential function $ e^{-x} $ is to substitute back for $ x $, completing the process.

Final Result

Math: $\int e^{-x} \,dx = -e^{-x} + C$

Explanation: The integration of the exponential function $ e^{-x} $ results in $-e^{-x} + C$, where $ C $ is the constant of integration. This detailed guide illustrates every algebraic step in the integration of the exponential function $ e^{-x} $, making it a comprehensive resource for anyone looking to understand this mathematical concept.

Conclusion

The integration of the exponential function $ e^{-x} $ is a fundamental concept in calculus. This step-by-step guide provides a clear and detailed explanation, ensuring a deep understanding of the integration of the exponential function $ e^{-x} $. Whether you are a student, educator, or math enthusiast, this guide to the integration of the exponential function $ e^{-x} $ offers valuable insights and clarity.

antiderivative of e^(-x)

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Integration of the Exponential Function \( e^{-x} \): Step-by-Step Guide