Problem 11 - Calculus for Biology and Medicine

Question

Evaluate the indefinite integral by making the given substitution. $$ \int x e^{-x^{2} / 2} d x, \text { with } u=-x^{2} / 2 $$

Step-by-Step Solution

Answer

\(-e^{-x^2/2} + C\)

Step 1: Identify the substitution

The substitution given is \( u = -\frac{x^2}{2} \). This means we are changing the variable of integration from \( x \) to \( u \).

Step 2: Differentiate the substitution

To find \( du \), differentiate \( u = -\frac{x^2}{2} \) with respect to \( x \): \( du = -x \, dx \).

Step 3: Rearrange the differential

From the differential equation obtained, solve for \( x \, dx \): \( x \, dx = -du \).

Step 4: Substitute into the integral

Substitute \( x \, dx = -du \) and \( u = -\frac{x^2}{2} \) into the original integral: \( \int x e^{-x^2/2} \, dx = \int e^{u} (-du) \).

Step 5: Simplify the integral

This simplifies to \( -\int e^{u} \, du \).

Step 6: Integrate with respect to \( u \)

The integral of \( e^{u} \) with respect to \( u \) is \( e^{u} \). Thus, the integral becomes \( -e^{u} + C \), where \( C \) is the constant of integration.

Step 7: Substitute back to the original variable

Substitute \( u = -\frac{x^2}{2} \) back into the expression: \( -e^{u} + C = -e^{-x^2/2} + C \).

Key Concepts

Substitution Method

The substitution method is a powerful tool in calculus, especially useful when dealing with integrals that don't immediately lend themselves to basic integration techniques. It works by transforming the integral into a simpler form that is easier to solve.
Essentially, you replace part of the original integral with a new variable. This involves:

• Choosing a substitution that simplifies the integral.
• Rewriting both the function and the differential in terms of the new variable.

In our exercise, the substitution is given as \( u = -\frac{x^2}{2} \). This transforms the complex exponential expression into a simpler exponential form. After substitution, the integral becomes more manageable, allowing us to apply basic integration rules smoothly.

Variable of Integration

The variable of integration is the variable with respect to which the function is being integrated. Identifying and changing the variable of integration is a crucial part of the substitution method.
Here's how it works:

• Initially, the integral is expressed in terms of a variable \( x \). In our problem, this variable corresponds to the \( x \) in \( \int x e^{-x^2/2} \, dx \).
• By substituting \( u = -\frac{x^2}{2} \), we aim to express the entire integral in terms of \( u \) instead of \( x \).
• This involves not only changing the function but also adjusting the differential element, which is done by finding \( du \) as \( du = -x \, dx \).

Once the substitution is complete, the variable of integration has been changed from \( x \) to \( u \), making the integral easier to evaluate.

Differentiation

Differentiation, in this context, plays a key role by enabling us to switch from the original variable to the substituted variable. It's used to find the derivative of the substitution function, allowing us to replace \( dx \) in the integral.
Consider these steps:

• Start with your substitution, \( u = -\frac{x^2}{2} \).
• By differentiating \( u \) with respect to \( x \), we get \( du = -x \, dx \). This relation tells us how changes in \( x \) translate to changes in \( u \).
• This differential relationship is essential for re-expressing the integral completely in terms of the new variable \( u \).

Differentiation simplifies the original integral problem into a format that's practical to solve, revealing the underlying structure and paving the way for a straightforward integration process.