Integrating e^x with Complex Expressions: A Step-by-Step Guide

When you encounter an expression involving e^x multiplied with a complex-looking integrand, the trick is often to apply a specific integration technique. This article will walk through the process of integrating e^x with a complex expression, specifically the form [x^3 - x^2 / (x^2 1)].

Introduction to the Problem

The integral in question is:

int e^x left[ frac{x^3 - x^2}{x^2 1} right] dx

The key challenge here is the complex-looking expression in the integrand. The standard result that helps in such cases is:

int e^x left[ f(x) f'(x) right] dx e^x f(x) C

Where one of the functions in the summand is a derivative of the other.

Breaking Down the Complex Expression

Let's break down the expression frac{x^3 - x^2}{x^2 1} into two functions, one of which is a derivative of the other. We achieve this through the method of partial fractions:

frac{x^3 - x^2}{x^2 1} frac{A x B}{x^2 1} frac{C}{x^2 1}

Multiplying through by (x^2 1) gives us:

x^3 - x^2 (A x B) x C

Equating Coefficients and Finding Constants

By comparing coefficients, we get the following system of equations:

A 1 B -1 C 0

Thus, the expression can be rewritten as:

frac{x^3 - x^2}{x^2 1} frac{x - 1}{x^2 1}

Further Decomposition and Integration

We further decompose the expression to make it look like a function and its derivative:

frac{x^3 - x^2}{x^2 1} frac{x}{x^2 1} - frac{x^2 - 1}{x^2 1}

And further:

frac{x^3 - x^2}{x^2 1} frac{x}{x^2 1} - frac{x}{x^2 1} frac{2 - 2x}{x^2 1}

Applying the Standard Integration Formula

Now that we have expressed the integrand in a form where one part is a function and the other is the derivative of the first, we can apply the standard integration formula:

Status

First, let’s integrate the two parts:

int e^x left[ frac{x}{x^2 1} frac{2 - 2x}{x^2 1} right] dx

Breaking it down, we get:

int e^x left[ frac{x}{x^2 1} - frac{x}{x^2 1} frac{2 - 2x}{x^2 1} right] dx int e^x left[ frac{x}{x^2 1} - frac{x}{x^2 1} frac{2 - 2x}{x^2 1} right] dx

This simplifies to:

int e^x left[ frac{x}{x^2 1} - frac{x}{x^2 1} frac{2 - 2x}{x^2 1} right] dx int e^x frac{x}{x^2 1} dx - int e^x frac{x}{x^2 1} dx int e^x frac{2 - 2x}{x^2 1} dx

Using the standard result, we integrate each term:

int e^x frac{x}{x^2 1} dx e^x frac{x}{x^2 1} C

int e^x frac{2 - 2x}{x^2 1} dx e^x frac{1}{x^2 1} C

Finally, combining the results, we get:

e^x frac{x}{x^2 1} - e^x frac{1}{x^2 1} C

Therefore, the final solution is:

e^x (frac{x}{x^2 1} - frac{1}{x^2 1}) C

Conclusion

This process demonstrates the power of recognizing patterns and the usefulness of the standard integration formula int e^x [f(x) f'(x)] dx e^x f(x) C. By breaking down complex expressions into simpler functions and their derivatives, integration becomes more manageable.