Question

In Problems 1-12, evaluate the given integral. $$ \int x e^{-5 x} d x $$

Short answer

The integral evaluates to \( -\frac{1}{5} x e^{-5x} - \frac{1}{25} e^{-5x} + C \).

Step-by-step solution

Identify the Method

The integral involves a product of a polynomial and an exponential function. This suggests that we use integration by parts, a technique that is particularly useful for such integrals. The formula for integration by parts is \( \int u \, dv = uv - \int v \, du \).

Choose \( u \) and \( dv \)

We choose \( u = x \) (a polynomial which simplifies upon differentiation) and \( dv = e^{-5x} \, dx \) (an exponential function that remains similar after integration).

Compute \( du \) and \( v \)

Differentiate \( u \) to find \( du \): \( du = dx \). Integrate \( dv \) to find \( v \): \( v = \int e^{-5x} \, dx = -\frac{1}{5} e^{-5x} \).

Apply the Integration by Parts Formula

Substitute \( u = x \), \( du = dx \), \( v = -\frac{1}{5} e^{-5x} \), and \( dv = e^{-5x} \, dx \) into the integration by parts formula: \[ \int x e^{-5x} \, dx = uv - \int v \, du = x \left(-\frac{1}{5} e^{-5x}\right) - \int \left(-\frac{1}{5} e^{-5x}\right) \, dx \]

Simplify and Solve the Remaining Integral

First simplify the expression: \[ = -\frac{1}{5} x e^{-5x} + \frac{1}{5} \int e^{-5x} \, dx \]Now solve the integral: \( \int e^{-5x} \, dx = -\frac{1}{5} e^{-5x} \)Thus, \[ \int x e^{-5x} \, dx = -\frac{1}{5} x e^{-5x} - \frac{1}{25} e^{-5x} + C \]

Write the Final Answer

Combine the terms to express the final answer of the integral: \[ \int x e^{-5x} \, dx = -\frac{1}{5} x e^{-5x} - \frac{1}{25} e^{-5x} + C \]where \( C \) is the constant of integration.

Key concepts

Exponential Functions

Exponential functions are mathematical functions that involve an exponential expression, typically in the form \( e^{ax} \) where \( e \) is Euler's number, approximately 2.71828, and \( a \) is a constant. These functions are powerful in modeling growth or decay processes in fields such as biology, economics, and physics. Exponential functions have unique properties that make them easy to integrate and differentiate. For example, the derivative of \( e^{ax} \) is \( ae^{ax} \) and its integral is \( \frac{1}{a} e^{ax} \), only differing by constant factors. This property is one of the reasons why exponential functions frequently appear in integration problems, especially when paired with other types of functions like polynomials. Applying integration by parts, as in the original exercise, often involves dealing with an exponential function so that its presence simplifies the integration process. Understanding how exponential functions behave under integration and differentiation is crucial for solving complex integrals.

Polynomial Integration

When integrating polynomials, you're dealing with expressions that involve sums of terms consisting of a variable raised to a power, such as \( x^n \). The rule for integrating a basic polynomial \( \int x^n \, dx \) is quite straightforward: \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. Polynomials have predictable behavior when differentiated or integrated, which is why they are often chosen as the function \( u \) in integration by parts. Differentiating a polynomial diminishes its degree, making the resulting integral simpler. In the context of integration by parts, choosing a polynomial for \( u \) is strategic, as it ensures the problem simplifies with successive differentiation. For example, in the original exercise, choosing \( u = x \) reduced the polynomial to a constant, making it easier to handle in subsequent steps.

Definite and Indefinite Integrals

Integrals come in two forms: definite and indefinite. Understanding the difference between them is essential for solving integral problems effectively.

• Definite Integrals: These integrals calculate the net area under a curve, typically over an interval \([a, b]\). The result is a number since it represents the cumulative area within specified limits. Mathematically, it is expressed as \( \int_{a}^{b} f(x) \, dx \), providing an exact value. For instance, definite integrals are used to find total quantities, such as distance or volume, based on variable rates.
• Indefinite Integrals: These are more general and represent a family of functions. The result is a function plus a constant \( C \), written as \( \int f(x) \, dx = F(x) + C \). This form indicates all possible antiderivatives of \( f(x) \), with \( C \) accounting for any constant shift. Indefinite integrals provide the base for solving more complex calculus problems, such as differential equations.

In the provided exercise, we calculated an indefinite integral, arriving at a general solution that included a constant \( C \). Understanding when to apply each type of integral is important for determining the right approach for solving mathematical problems.