Problem 60 - Calculus Early Transcendentals

Question

Miscellaneous integrals Evaluate the following integrals. \(\int_{0}^{5} 5^{5 x} d x\)

Step-by-Step Solution

Answer

Answer: The expression for the definite integral of 5^(5x) from x = 0 to x = 5 is (5^(25) - 1) / ln(5^5).

Step 1: Find the antiderivative of the integrand

To find the antiderivative of \(5^{5x}\), we can recall that the derivative of \(a^{x}\) (where \(a\) is a constant) is \(a^{x} \ln a\). Therefore, the antiderivative of \(5^{5 x}\) with respect to x is given by: $$\frac{5^{5x}}{\ln{5^5}}\, + C $$

Step 2: Apply the limits of integration

Now, let's further evaluate the integral by applying the limits of integration using the Fundamental Theorem of Calculus: $$\int_{0}^{5} 5^{5 x} d x = \left[\frac{5^{5x}}{\ln{5^5}}\right]_0^5$$

Step 3: Evaluate the antiderivative at the upper and lower limits

Evaluate the antiderivative at the upper limit (x = 5) and the lower limit (x = 0) and subtract the result of the lower limit from the result of the upper limit: $$\frac{5^{5(5)}}{\ln{5^5}} - \frac{5^{5(0)}}{\ln{5^5}} = \frac{5^{25}}{\ln{5^5}} - \frac{5^0}{\ln{5^5}}$$

Step 4: Simplify and compute the answer

Now let's simplify and compute the final answer: $$\frac{5^{25}}{\ln{5^5}} - \frac{1}{\ln{5^5}} = \frac{5^{25} - 1}{\ln{5^5}}$$ The value of this integral may not be easily simplified by hand, but it can be computed using a calculator or software to obtain a numerical value. The final answer is: $$\int_{0}^{5} 5^{5x} dx = \frac{5^{25} - 1}{\ln{5^5}}$$

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, two of the main operations in calculus. It states that if you have a continuous function and you take its indefinite integral, you can differentiate it to get back to the original function. Or, in simpler terms, integration and differentiation are inverse processes.
The theorem consists of two parts:

• The first part states that if a function is continuous on a closed interval \([a, b]\), then the function has an antiderivative over that interval. This means you can calculate the integral from \([a, b]\) using this antiderivative.
• The second part lets us evaluate a definite integral by subtracting the values of an antiderivative at the upper and lower bounds of the interval.

In the original step-by-step solution, we used this theorem to evaluate the integral of the exponential function. We found an antiderivative first, then evaluated it at specific limits to get our solution. This approach is typical for definite integrals, transforming an otherwise complex calculation into a more manageable subtraction problem.

Exponential Functions

Exponential functions are a type of mathematical expression often written in the form \(a^x\), where \(a\) is a constant and \(x\) is the exponent. In our specific problem, we tackled the exponential function \(5^{5x}\).
One of the key properties of exponential functions is that they grow rapidly as \(x\) increases, especially when the base (in this case, 5) is greater than 1. This growth behavior makes them widely applicable in fields such as finance, science, and engineering.
When differentiating, the derivative of \(a^x\) results in \(a^x\ln a\). This property of exponential functions makes them relatively straightforward to work with when finding integrals or derivatives.
In integration, we're often interested in finding the antiderivative, which is the reverse process of differentiation. For \(5^{5x}\), the antiderivative involves dividing by the natural logarithm of the base raised to the original exponent, leading us to \(\frac{5^{5x}}{\ln{5^5}}\). This step is crucial for solving any integral involving exponential functions.

Integration Techniques

Integration techniques are methods used to solve integrals, and understanding these techniques is crucial for tackling various problems in calculus.
For exponential functions, the integral often involves using specific formulas and recognizing patterns. The problem involving \(5^{5x}\) employs the technique of using known antiderivative forms of exponential functions.
Here are some basic techniques that often come in handy when dealing with integration:

• Antiderivatives: Recognizing the antiderivative of a function can simplify the process. For exponentials like \(a^x\), applying the antiderivative formula directly helps.
• Substitution: Useful when the integrand is a composite function, you replace a part of the function with a new variable to simplify integration.
• Integration by Parts: Particularly for products of functions, it transforms the integral into parts that are easier to manage.

Each technique requires practice to identify which approach to use in varying scenarios. Calculators and software can also compute complex integrals, especially those without closed-form solutions. \(\int_{0}^{5} 5^{5x} dx\) was straightforward with direct application of known formulas, saving us from more complex methods.