Question

State whether you would use integration by parts to evaluate the integral. If so, identify u and dv. If not, describe the technique used to perform the integration without actually doing the problem. $$ \int x e^{x^{2}-3} d x $$

Short answer

Use substitution, not integration by parts, with \( u = x^2 - 3 \).

Step-by-step solution

Identify the Type of Integral

Look at the integral \( \int x e^{x^{2}-3} \, dx \). This integral involves a product of a polynomial \( x \) and an exponential function of the form \( e^{x^{2}-3} \).

Consider a Suitable Integration Technique

Given the structure of the exponential function \( e^{x^2-3} \), you may consider if substitution could simplify the integral. By setting a substitution based on the exponent, it could simplify the integration.

Evaluate Substitution

Let \( u = x^2 - 3 \), then \( du = 2x \, dx \), or \( \frac{1}{2} du = x \, dx \). Substitute to simplify the integral to \( \int \frac{1}{2} e^u \, du \).

Determine the Integration Method

Since substitution simplifies the integral, integration by parts is not necessary in this case. Thus, the integral can be evaluated using the substitution method.

Key concepts

Integration by Substitution

Integration by substitution, also known as u-substitution, is a technique to simplify complex integrals. This method is particularly useful when the integrand (the function being integrated) involves a composition of functions. If you can identify a part of the integrand whose derivative appears elsewhere in the integral, substitution can be considered effective.

For example, if the integrand contains a polynomial inside another function, as in exponentials, a substitution can help reduce the complexity:

• Choose a new variable, say, \( u \), to replace a part of the integrand. In our case, \( u = x^2 - 3 \).
• Differentiate \( u \) to find \( du \), which gives \( du = 2x \, dx \).
• Substitute \( x \, dx \) in the integral by \( \frac{1}{2} du \).
• The integral now becomes a simpler form \( \int \frac{1}{2} e^u \, du \).

Once substitution reshapes the integral to a simpler form, solving becomes more straightforward. Integration becomes a standard process involving simpler functions.

Integration by Parts

Integration by parts is another important technique, generally used when the integrand is a product of functions that do not lend themselves easily to substitution. It is based on the formula: \[\int u \, dv = uv - \int v \, du\]
This technique is ideal when dealing with integrals involving polynomials multiplied by logarithmic or trigonometric functions.

If substitution doesn't directly simplify the integral, integration by parts may be the preferred choice. To apply this method, one would typically:

• Identify components of the integrand to assign as \( u \) and \( dv \).
• Differentiate \( u \) to find \( du \) and integrate \( dv \) to find \( v \).
• Apply the integration by parts formula.

However, in the provided integral \( \int x e^{x^2-3} \, dx \), substitution effectively simplifies it, thus obviating the need for integration by parts.

Polynomial and Exponential Functions

Understanding the behavior of polynomial and exponential functions is crucial when dealing with calculus problems. Polynomials include expressions like powers of \( x \), and they are straightforward to integrate when standalone.
However, when involved in compositions with exponential functions—such as in \( e^{x^2-3} \)—these require more attention.

Exponential functions, particularly those involving complicated exponents, can often be simplified using substitution. This occurs because their derivatives are inherently related to their original expressions, making them smoother to handle mathematically.
For example:

• Exponential functions like \( e^u \) remain unchanged in form upon differentiation and integration, simplifying processes greatly.
• When combined with polynomials, a focused approach like substitution can elegantly untangle the integration process.

Thus, recognizing the structure of the function helps in choosing the correct technique to solve integrals efficiently.