Chapter 1: Problem 10

Evaluate the definite integrals by expanding the integrand in a Maclaurin series. \(\int_{0}^{1} \frac{e^{x}-1}{x} d x\)

Short Answer

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Step by step solution

Recall the Maclaurin series for the exponent function

The Maclaurin series for the exponential function is given by

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Definite Integrals

A definite integral represents the area under a curve from one point to another. It has two limits: an upper limit and a lower limit. In our exercise, the definite integral is written as \(\text{@int_0^1({(e^x-1)/x})dx}@\). Here, the limits are from 0 to 1.
This integral calculates the total accumulation or area of the function \(\frac{e^x-1}{x}\) between 0 and 1.
When evaluating definite integrals, it’s important to understand the properties of the function and the behavior near the limits. In this case, using the Maclaurin series makes calculations easier.

Maclaurin Series

The Maclaurin series is a type of Taylor series expanded at 0. It provides a polynomial approximation of a function.
For the exponential function \( e^x \), the Maclaurin series is written as: \[ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots\]
To solve the given integral, we need to replace \(e^x\) with its Maclaurin series: \[ \frac{e^x - 1}{x} = \frac{(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots) - 1}{x}\]
Simplifying gives: \[ 1 + \frac{x}{2!} + \frac{x^2}{3!} + \ldots\] By integrating this series term-by-term, we can then evaluate the definite integral.

Exponential Function

The exponential function \(e^x\) is fundamental in mathematics, especially in calculus and differential equations. It is defined by its unique property, where the derivative of \(e^x\) is \(e^x\) itself.
In series form, it is represented using the Maclaurin series as: \[ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots\]
Understanding this function and its series representation allows us to handle complex integrals, like in our exercise. The series representation makes it easier to approximate and manipulate the function for integration.

Integration Techniques

Different techniques can be used for integration, depending on the function we are dealing with. The method used in our example involves expanding the integrand using the Maclaurin series.
Here’s what we did step-by-step:

Expanded \(e^x\) into its Maclaurin series.
Substituted this series into the original integral.
Integrated term-by-term.
Evaluated the definite integral by finding the anti-derivative and then applying the limits from 0 to 1.

This method is particularly useful for functions where direct integration is complex. Techniques like substitution, integration by parts, and partial fractions can also be useful, depending on the nature of the integral.

Evaluate the definite integrals by expanding the integrand in a Maclaurin series. \(\int_{0}^{1} \frac{e^{x}-1}{x} d x\)

Short Answer

Step by step solution

Recall the Maclaurin series for the exponent function

step title

Key Concepts

Definite Integrals

Maclaurin Series

Exponential Function

Integration Techniques

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