Question

Compute the definite integrals. Use a graphing utility to confirm your answers. $$ \int_{0}^{1} x 5^{x} d x \text { (Express the answer using five significant digits.) } $$

Short answer

The approximate value of the integral is 0.68605.

Step-by-step solution

Identify the Components

We need to compute the definite integral \( \int_{0}^{1} x 5^{x} \, dx \). Here, \( x 5^x \) is the function to be integrated from 0 to 1.

Choose the Integration Technique

The function \( x 5^x \) involves a polynomial and an exponential function. This suggests using integration by parts, which is written as \( \int u \, dv = uv - \int v \, du \).

Apply Integration by Parts

Let \( u = x \) and \( dv = 5^x \, dx \). Compute \( du = dx \) and integrate \( dv \) to get \( v = \int 5^x \, dx = \frac{5^x}{\ln(5)} \). Applying integration by parts yields:\[ \int x 5^x \, dx = x \frac{5^x}{\ln(5)} - \int \frac{5^x}{\ln(5)} \, dx. \]

Integrate \( \int \frac{5^x}{\ln(5)} \, dx \)

The integral \( \int \frac{5^x}{\ln(5)} \, dx \) can be computed as \( \frac{5^x}{(\ln(5))^2} \). Substituting back, we have:\[ \int x 5^x \, dx = x \frac{5^x}{\ln(5)} - \frac{5^x}{(\ln(5))^2} + C. \]

Evaluate the Definite Integral

Evaluate from 0 to 1:\[ \left[ x \frac{5^x}{\ln(5)} - \frac{5^x}{(\ln(5))^2} \right]_0^1. \]Compute at each bound:\[ \left( 1 \cdot \frac{5^1}{\ln(5)} - \frac{5^1}{(\ln(5))^2} \right) - \left( 0 \cdot \frac{5^0}{\ln(5)} - \frac{5^0}{(\ln(5))^2} \right). \]

Simplify and Compute the Numeric Result

Substitute and compute the values:\[ \left( \frac{5}{\ln(5)} - \frac{5}{(\ln(5))^2} \right) - \left( 0 - \frac{1}{(\ln(5))^2} \right). \]Simplify this to get a numerical value. Using a calculator, approximate the result to five significant digits.

Verification with Graphing Utility

A graphing utility can confirm the definite integral calculation. Input the function \( x 5^x \) and compute \( \int_{0}^{1} \) to verify. Confirm that the result matches the computed value from previous steps.

Key concepts

Definite Integral

A definite integral represents the area under a curve for a specific interval. In this exercise, we calculate the area from 0 to 1 for the function \(x 5^x\). The definite integral is expressed as \(\int_{a}^{b} f(x) \, dx\). Here, \(a\) and \(b\) are the limits of integration, which for our problem are 0 and 1 respectively. The function \(f(x)\) is \(x 5^x\), a combination of polynomial and exponential terms. To find the definite integral, we follow these steps:

• Calculate the indefinite integral (antiderivative) of the function.
• Evaluate the antiderivative at the upper limit \(b\) and lower limit \(a\).
• Subtract the lower limit value from the upper limit value.

This process gives you the precise area under the curve from \(x = 0\) to \(x = 1\). In practical applications, definite integrals help compute quantities like area, volume, and other real-world applications where limits are crucial.

Exponential Functions

Exponential functions are crucial in this problem as they dictate the growth rate of our function. An exponential function takes the shape \(a^x\), where \(a\) is a constant base and \(x\) is the exponent. In the given integral, the exponential component is \(5^x\), which grows rapidly as \(x\) increases, significantly affecting the shape of the curve.Exponential functions have the unique property that their rate of change is proportional to their current value. This characteristic makes them appear frequently in modeling population growth, radioactive decay, and other natural phenomena.
When integrating \(5^x\), a key step is recognizing how it transforms. The integral of \(5^x\) is \(\frac{5^x}{\ln(5)}\). Here we use the logarithm of the base \(\ln(5)\) for proper scaling, allowing the function to be integrated accurately. Understanding these properties helps in dealing with exponential terms within an integral.

Integration Techniques

Integration by parts is the preferred technique for our integral \(\int x 5^x \, dx\). This method simplifies the process of integrating products of functions by breaking them down into parts. It's especially useful when you have one easy-to-differentiate and one easy-to-integrate term. The formula for integration by parts is \(\int u \, dv = uv - \int v \, du\) and involves the following steps:

• Assign \(u = x\) and \(dv = 5^x \, dx\) based on simplicity for differentiation and integration respectively.
• Differentiate \(u\) to get \(du\) and integrate \(dv\) to find \(v\).
• Substitute into the formula, which leads to parts that are easier to solve.

Once organized, simplify and evaluate over the given limits. This strategic approach avoids complex integrations by methodically simplifying each component. It's a powerful technique for tackling definite integrals involving polynomial and exponential functions, as shown in this exercise.