Problem 42 - Calculus for Biology and Medicine

Question

In Problems \(39-48\), first make an appropriate substitution and then use integration by parts to evaluate each integral. $$ \int x^{5} e^{x^{2}} d x $$

Step-by-Step Solution

Answer

\(\frac{1}{2} x^4 e^{x^2} - x^2 e^{x^2} + e^{x^2} + C\)

Step 1: Identify Suitable Substitution

To begin, we identify a substitution that simplifies the integral. Notice that the exponential function involves \(x^2\), so a natural substitution is to let \(u = x^2\). Consequently, \(du = 2x \, dx\) or \(dx = \frac{du}{2x}\).

Step 2: Apply Substitution

Substitute \(u = x^2\) and rearrange the differential: \(dx = \frac{du}{2x}\). The integral becomes \(\int x^5 e^{x^2} \, dx = \int \frac{x^5}{x} e^u \left(\frac{du}{2}\right) = \int \frac{x^4}{2} e^u \, du\).

Step 3: Simplify the Expression

Further simplify the integral: \(\frac{1}{2} \int x^4 e^u \, du\). With \(x^2 = u\), it follows that \(x^4 = u^2\). Substitute \(x^4 = u^2\) to get \(\frac{1}{2} \int u^2 e^u \, du\).

Step 4: Integration by Parts Setup

Integration by parts is needed: \(\int v \, dw = uv - \int u \, dv\). Choose \(v = u^2\) and \(dw = e^u \, du\). Then, \(dv = 2u \, du\) and \(w = e^u\).

Step 5: Apply Integration by Parts

Substitute into the integration by parts formula: \(\int u^2 e^u \, du = u^2 e^u - \int 2u e^u \, du\).

Step 6: Recursive Integration by Parts for Remaining Integral

For \(\int 2u e^u \, du\), apply integration by parts again: let \(v = u\) and \(dw = e^u \, du\), thus \(dv = du\) and \(w = e^u\). Use the formula to evaluate \(\int 2u e^u \, du = 2(ue^u - \int e^u \, du) = 2(ue^u - e^u) = 2u e^u - 2e^u\).

Step 7: Combine Results

Substitute back the result into the original integration by parts: \(u^2 e^u - (2u e^u - 2e^u) = u^2 e^u - 2ue^u + 2e^u\).

Step 8: Replace Substitution

Switch back to the variable x. Since \(u = x^2\), the expression \(x^4 e^{x^2} - 2x^2 e^{x^2} + 2e^{x^2}\) results. Therefore, the integral is \(\frac{1}{2} (x^4 e^{x^2} - 2x^2 e^{x^2} + 2e^{x^2})\).

Step 9: Final Expression

The final result is \(\frac{1}{2} x^4 e^{x^2} - x^2 e^{x^2} + e^{x^2} + C\), where C is the constant of integration.

Key Concepts

Substitution Method

The substitution method is a powerful tool for solving integrals. It's like changing the variable to simplify computation. Consider the integral \( \int x^5 e^{x^2} \, dx \). The goal is to identify a substitution that can simplify this integral. In this context, notice the presence of \(x^2\) in the exponential function. This hints that we might simplify the integral by letting \( u = x^2 \). Consequently, the derivative of \( u \) with respect to \( x \) is \( du = 2x \, dx \). Rearranging gives us \( dx = \frac{du}{2x} \).

By substituting \( u \) in for \( x^2 \) and transforming \( dx \), the integral \( \int x^5 e^{x^2} \, dx \) can now be rewritten into a simpler form: \( \int \frac{x^4}{2} e^u \, du \), where \( x^4 = u^2 \). This substitution simplifies the integral into a form that can be more easily tackled, and it serves as the first step in breaking down complex integrals.

Integration by Parts

Integration by parts is an essential technique in calculus for solving integrals involving products of functions. It's based on the product rule for differentiation and is given by the formula:

• \( \int v \, dw = uv - \int u \, dv \)

In our problem, after making the substitution, we have \( \int u^2 e^u \, du \). Here, integration by parts allows us to dismantle this expression further. Choose \( v = u^2 \) and \( dw = e^u \, du \), leading to \( dv = 2u \, du \) and \( w = e^u \). Applying these choices to the integration by parts formula results in:

• \( \int u^2 e^u \, du = u^2 e^u - \int 2u e^u \, du \)

This again leaves us with a new integral, \( \int 2u e^u \, du \), to solve using the same method. Ultimately, integration by parts simplifies handling products of polynomial and exponential functions.

Definite Integrals

Definite integrals involve calculating the area under a curve between two points and are usually evaluated with specified limits. However, in this exercise, we focus on indefinite integrals, which include integrating without specific limits and result in an expression that includes a constant of integration \( C \).

After performing integration by parts twice, we are left with a final expression that can be written out fully as \( \frac{1}{2} (x^4 e^{x^2} - 2x^2 e^{x^2} + 2e^{x^2}) \), where the variable substitution is reverted to \( x \) instead of \( u \).

The result of the indefinite integral captures the family of all possible antiderivatives of the original function \( \int x^5 e^{x^2} \, dx \). Adding the constant \( C \) accounts for the fact that any constant could achieve the same derivative, emphasizing the general solution over specific numerical bounds.