Calculate The Integral 0 Ln 2 Ex Dx 1 E2x

This guide explains how to calculate the definite integral of e^x from 0 to ln(2), which equals 1/e². We'll cover the mathematical process, provide a calculator, and discuss practical applications.

What is this integral?

The integral ∫ from 0 to ln(2) of e^x dx represents the area under the curve of the exponential function e^x between x=0 and x=ln(2). This is a fundamental calculus problem that demonstrates the relationship between exponential functions and their antiderivatives.

∫₀^ln(2) e^x dx = e^x evaluated from 0 to ln(2) = e^ln(2) - e⁰ = 2 - 1 = 1

This result shows that the area under e^x from 0 to ln(2) is exactly 1. The exponential function grows so rapidly that the area under it between these bounds is a simple integer value.

How to calculate this integral

Calculating this integral involves finding the antiderivative of e^x and evaluating it at the bounds. Here's the step-by-step process:

Identify the antiderivative of e^x, which is also e^x.
Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper bound (ln(2)) and subtracting its value at the lower bound (0).
Simplify the expression using the property that e^ln(a) = a.

∫ e^x dx = e^x + C

∫_a^b e^x dx = e^b - e^a

For our specific integral:

∫₀^ln(2) e^x dx = e^ln(2) - e⁰ = 2 - 1 = 1

This calculation shows that the area under e^x from 0 to ln(2) is exactly 1, demonstrating the precise relationship between the exponential function and its antiderivative.

Example calculation

Let's work through a concrete example to see how this integral is calculated in practice.

Example: Calculate ∫ from 0 to ln(2) of e^x dx

Find the antiderivative of e^x, which is e^x.
Evaluate e^x at the upper bound (ln(2)): e^ln(2) = 2.
Evaluate e^x at the lower bound (0): e⁰ = 1.
Subtract the lower evaluation from the upper evaluation: 2 - 1 = 1.

The result is 1, which matches our earlier calculation. This example demonstrates how the properties of exponential functions and logarithms simplify the integral calculation.

Interpreting the result

The result of 1 for this integral has several important implications:

It shows that the area under e^x from 0 to ln(2) is exactly 1, demonstrating the precise relationship between exponential growth and integration.
This result is counterintuitive at first glance because e^x grows rapidly, but the area under it between these specific bounds is a simple integer.
The calculation relies on the property that e^ln(a) = a, which is fundamental to understanding the relationship between exponential and logarithmic functions.

Understanding this integral helps in more advanced calculus topics and has practical applications in probability, statistics, and physics where exponential functions are commonly used.

FAQ

What is the antiderivative of e^x?: The antiderivative of e^x is also e^x, plus a constant of integration. This makes e^x its own antiderivative.
Why is the result of this integral 1?: The result is 1 because e^ln(2) equals 2, and e⁰ equals 1. Subtracting these gives 2 - 1 = 1.
What are practical applications of this integral?: This integral appears in probability theory, where it's used to calculate probabilities for exponential distributions. It's also fundamental in understanding the relationship between exponential and logarithmic functions.
Can I use this calculator for other integrals?: This calculator is specifically designed for calculating ∫ from 0 to ln(2) of e^x dx. For other integrals, you would need a different calculator or mathematical software.
Is this result always true for any exponential function?: No, this specific result only holds for the integral of e^x from 0 to ln(2). Other exponential functions or different bounds would yield different results.