Problem 61 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int x e^{-x^{2} / 2} d x $$

Step-by-Step Solution

Answer

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

Step 1: Recognize the Integral Form

Notice that the integration can be solved by using substitution. The integral is of the form \( \int u \, e^{u} \, du \), which suggests substitution might be useful.

Step 2: Choose the Appropriate Substitution

Let \( u = -\frac{x^2}{2} \). Then the derivative \( du = -x \, dx \). This means \( dx = -\frac{du}{x} \). Substitute \( u \) into the integral.

Step 3: Rewrite the Integral with Substitution

The integral becomes: \[\int x e^{-x^2/2} \, dx = \int e^{u} \, \left(-\frac{du}{x}\right) \, x.\]The \( x \) from \( x \, dx \) cancels out, giving:\[-\int e^{u} \, du.\]

Step 4: Integrate with Respect to \( u \)

The integral \( \int e^{u} \, du \) is simple and evaluates to \( e^{u} + C \), where \( C \) is the constant of integration. Since there is a negative sign, it becomes \[-e^{u} + C.\]

Step 5: Substitute Back for \( u \)

Substitute the value of \( u \) back into the solution:\[-e^{-x^2/2} + C.\]

Key Concepts

Integration by Substitution

Integration by substitution is a powerful technique often used to simplify the integration of complex functions. It's akin to reversing the chain rule in differentiation. You begin by identifying a part of the integrand (the function being integrated) that, when substituted with a single variable, can make the integral simpler. For this exercise, you recognize

• that the expression \(-\frac{x^2}{2}\) forms a more straightforward derivative and can be denoted by a substitution variable like \(u\).

The process is crucial, as it transforms an intimidating integral into one that is more manageable. You convert the expression using \(u\), effectively changing the variable of integration from \(x\) to \(u\), which can then be integrated more simply. After calculating the integral in terms of \(u\), you revert or "substitute back" to express the answer in terms of the original variable, which ensures your solution aligns with the function given in the problem statement.

Exponential Functions

Exponential functions, which take the form \(e^x\), are frequently encountered in calculus due to their unique properties. They are distinguished by their growth rate functionally dependant on the constant 'e', approximately equal to 2.71828. This intrinsic property means that the derivative and integral of \(e^u\) find themselves elegantly similar:

• The integral \(\int e^{u} \, du = e^u + C\) remains one of the neat forms thanks to its nature.

Understanding exponential functions is crucial because they often appear in mathematical modeling and complex calculations, much like in population growth and radioactive decay. In our exercise, recognizing \(e^{-x^2/2}\) helps us identify the function simplification needed. After the substitution for \(u\), the integration takes advantage of this simplified form of the exponential function by expressing it as a direct integral of \(e^u\), leading smoothly to a solution.

Constant of Integration

A cornerstone of indefinite integrals is the constant of integration, denoted by \(C\). When you compute an indefinite integral, you're actually identifying a family of functions. This constant represents an infinite number of vertical translations of the antiderivative. It is crucial:

• because it ensures your solution encompasses all possible functions that could have produced the original integrand when differentiated.

In the solution provided, after we reintegrate using the simpler form found in exponential functions and substitution, the constant \(C\) is appended to the integrated expression \(-e^{-x^2/2}\). This ensures that our integral solution \(-e^{-x^2/2} + C\) is general enough to include any arbitrary constant that could have been part of the original function prior to differentiation.