Problem 12 - Calculus for Biology and Medicine

Question

In Problems 1-16, evaluate each indefinite integral by making the given substitution. $$ \int x e^{1-2 x^{2}} d x, \text { with } u=1-2 x^{2} $$

Step-by-Step Solution

Answer

The integral evaluates to \( -\frac{1}{4} e^{1-2x^2} + C \).

Step 1: Identify the Given Substitution

The substitution provided in the problem is \( u = 1 - 2x^2 \). This substitution will simplify the integral.

Step 2: Differentiate the Substitution

Differentiate both sides of the substitution \( u = 1 - 2x^2 \) with respect to \( x \). This gives \( \frac{du}{dx} = -4x \), or equivalently, \( du = -4x \, dx \).

Step 3: Solve for the Expression with dx

Since we need \( x\,dx \) in the integral, we rearrange \( du = -4x \, dx \) to solve for \( x\,dx \): \( x\,dx = -\frac{1}{4}du \).

Step 4: Substitute into the Integral

Substitute \( u = 1 - 2x^2 \) and \( x\,dx = -\frac{1}{4}du \) into the integral. The integral becomes \( \int x e^{1-2x^2} \, dx = \int -\frac{1}{4} e^u du \).

Step 5: Integrate with Respect to u

Integrate \( \int -\frac{1}{4} e^u \, du \). This results in \( -\frac{1}{4} \int e^u \, du = -\frac{1}{4} e^u + C \), where \( C \) is the integration constant.

Step 6: Substitute Back in Terms of x

Replace \( u \) with \( 1 - 2x^2 \) to express the integral in terms of \( x \). The final result is \( -\frac{1}{4} e^{1-2x^2} + C \).

Key Concepts

Substitution Method

The substitution method is a helpful technique in calculus for simplifying integrals by changing variables. This method makes integration more manageable, especially when dealing with complex forms or expressions.

In this problem, to tackle the integral \( \int x e^{1-2x^2} dx \), a substitution \( u = 1 - 2x^2 \) is provided. This substitution transforms the integral into a simpler form that is easier to evaluate.

The process involves a few critical steps:

• Identify the substitution variable \( u \) and express it in terms of \( x \).
• Differentiate the substitution to find \( du \) in terms of \( dx \) and \( x \).
• Solve for the necessary terms in the original integral, in this case, \( x \, dx \), using the derivative equation.
• Substitute both \( u \) and \( x \, dx \) back into the integral to transform it.

With the integral now in terms of \( u \), it becomes \( -\frac{1}{4} \int e^u \, du \). The substitution method effectively turns the task of integrating a complex expression into a simpler, more straightforward problem.

Integration

Integration is the process of finding the integral of a function, which can be thought of as the "reverse" of differentiation. For indefinite integrals, the solution includes a constant of integration, denoted as \( C \).

In our example, after performing the substitution and simplifying the original integral, we are tasked to integrate \( -\frac{1}{4} e^u \, du \). Because the exponential function \( e^u \) is straightforward, its integral is itself plus a constant.

This yields:

• \( \int e^u \, du = e^u + C \)
• Thus, \( -\frac{1}{4} \int e^u \, du = -\frac{1}{4} e^u + C \)

Integrating exponential functions is a common task in calculus due to their vast applications and how frequently they appear in various fields of study. Once integrated in terms of \( u \), the next step involves reverting back to the original variable \( x \). This is where we substitute back \( u = 1 - 2x^2 \) to express the final answer in terms of \( x \).

Calculus

Calculus is the mathematical study of continuous change and is split mainly into two branches: differentiation and integration. Indefinite integrals, like the one we’re solving, are a fundamental part of calculus representing the accumulation of quantities, such as area under a curve.

The process of solving \( \int x e^{1-2x^2} dx \) is a perfect illustration of how calculus transforms and manipulates equations to unravel complex problems. Key ideas include:

• The change of variables through substitution helps simplify integration tasks.
• Integration itself is a technique to find total accumulations, such as mass, area, or volume.
• Each integral is associated with a constant \( C \), denoting the family of functions that differentiate to the same original function under different scenarios.

In essence, calculus is foundational for understanding many natural phenomena, crafting solutions to intricate problems, and driving technological advancements. The substitution method within integration represents a small yet crucial portion of this extensive field.