Problem 48 - Calculus of a Single Variable

Question

Comparing Integration Problems In Exercises \(47-50\) , determine which of the integrals can be found using the basic integration formulas you have studied so far in the text. $$ \begin{array}{l}{\text { (a) } \int e^{x^{2}} d x} \\ {\text { (b) } \int x e^{x^{2}} d x} \\ {\text { (c) } \int \frac{1}{x^{2}} e^{1 / x} d x}\end{array} $$

Step-by-Step Solution

Answer

Among the given integrals (a), (b), and (c), only integral (b) \( \int x e^{x^{2}} dx \) can be solved using the basic integration formulas studied so far in the text.

Step 1: Inspect Integral (a)

Inspect \( \int e^{x^{2}} dx \). This integral cannot be solved using basic integration formulas as there is no formula available for integrating function of the form \( \int e^{f(x)} dx \) where \( f(x) \) is a polynomial of degree greater than 1.

Step 2: Inspect Integral (b)

Inspect \( \int x e^{x^{2}} dx \). This is a standard problem of integration by substitution. If we let \( u = x^2 \), then \( du = 2x dx \). This substitution can convert the integral into the simpler form \( \frac{1}{2} \int e^u du \), which is solvable.

Step 3: Inspect Integral (c)

Inspect \( \int \frac{1}{x^{2}} e^{1 / x} dx \). This integral also cannot be solved using basic integration formulas as functions in the form \( \int e^{f(x)} dx \) where \( f(x) \) is \( 1/x \), or a rational function, do not have a basic anti-derivative.

Key Concepts

Basic Integration Formulas

Basic integration formulas are the foundation of calculus. These formulas allow us to find antiderivatives for many common functions easily.
When we see an integral, the first thing we should do is check if we can use a basic formula to solve it.Some of the most commonly used basic integration formulas encompass:

• The integral of a power of x: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \)
• The integral of the exponential function: \( \int e^x \, dx = e^x + C \)
• The integral of the sine function: \( \int \sin x \, dx = -\cos x + C \)
• The integral of the cosine function: \( \int \cos x \, dx = \sin x + C \)

When examining the problem's integral (a) \( \int e^{x^2} \, dx \), it's clear we do not have a straightforward basic formula to apply, since the exponent is a polynomial of degree greater than 1, requiring more advanced techniques than basic formulas.

Integration by Substitution

Integration by substitution is a method used when an integral takes a more complex form. This technique simplifies the process by changing the variable of integration. It is particularly useful when dealing with a composite function.The technique can be summarized as follows:

• Choose a substitution \( u = f(x) \), where \( f(x) \) is part of the integral and simplifies it when substituted.
• Differentiate \( u \), giving \( du = f'(x) \, dx \).
• Replace all instances of \( x \) and \( dx \) in the integral with \( u \) and \( du \) respectively. This often results in a simpler integral to solve.
• After integration, substitute back the original variable \( x \).

For integral (b) \( \int x e^{x^2} \, dx \), by letting \( u = x^2 \), we find that \( du = 2x \, dx \). This substitution converts the integral to \( \frac{1}{2} \int e^u \, du \), which is straightforward to integrate.

Polynomial Integration

Polynomial integration refers to the process of integrating functions composed of polynomials, which is generally straightforward with basic integration formulas. To integrate a polynomial, apply the power rule formula, which states:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), as long as \( n eq -1 \)

When dealing with more complex scenarios, such as having a polynomial within another function, substitution or other methods may be required. Although integral \( \int e^{x^2} \, dx \) involves a polynomial in the exponent, it does not simplify directly with polynomial integration or basic formulas, necessitating different techniques.

Rational Function Integration

Rational function integration involves finding the antiderivative of functions that are ratios of polynomials, which are often expressed in the form \( \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials.Several techniques can be applied, depending on the complexity of the rational function:

• **Partial Fraction Decomposition**: Breaking down a complex rational expression into simpler components that are easier to integrate.
• **Substitution**: Sometimes a simple change of variable can simplify the integral.

For integral (c) \( \int \frac{1}{x^2} e^{1/x} \, dx \), neither partial fractions nor a straightforward substitution is immediately evident. The presence of \( e^{1/x} \) in the integrand complicates the matter beyond standard rational integration techniques, necessitating more advanced approaches or tools for a solution.