Problem 24 - Essential Calculus

Question

Evaluate the definite integral of the transcendental function. Use a graphing utility to verify your result. $$ \int_{0}^{3}\left(t-5^{t}\right) d t $$

Step-by-Step Solution

Answer

The definite integral of \( t-5^{t} \) from 0 to 3 equals \( \frac{9}{2} - \frac{5^{3}}{\ln 5} + 1 \)

Step 1: Break down into simpler integrals

The integral can be broken down into two simpler ones: \[\int_{0}^{3}\left(t-5^{t}\right) d t = \int_{0}^{3} t \, dt - \int_{0}^{3} 5^{t} \, dt \]

Step 2: Evaluate \( \int_{0}^{3} t \, dt \)

The integral of \( t \) from 0 to 3 is obtained directly by applying the power rule:\[\frac{t^{2}}{2}\Bigg|_{0}^{3} = \frac{9}{2} \]

Step 3: Evaluate \( \int_{0}^{3} 5^{t} \, dt \)

The integral of \( 5^{t} \) is obtained using a special integral rule for \( a^{t} \):\[\int a^{t} \, dt = \frac{a^{t}}{\ln a} \Bigg|_{0}^{3} = \frac{5^{3}}{\ln 5} - 1\]

Step 4: Add the integral results

The total integral is obtained by adding the results of the separate integrals:\[\int_{0}^{3}\left(t-5^{t}\right) d t = \frac{9}{2} - \frac{5^{3}}{\ln 5} + 1\]

Step 5: Verify Result with Graphing Utility

Verify the correctness of the calculated integral using a graphing utility by comparing the performed area under the function \( t - 5^{t} \) from 0 to 3.

Key Concepts

Transcendental Function

A transcendental function is a kind of function that goes beyond algebraic functions. These cannot be expressed just with a finite number of algebraic operations (such as addition, multiplication, or division). Famous examples of transcendental functions include the exponential, logarithmic, and trigonometric functions.
In the context of the given exercise, the term \(5^t\) represents a transcendental part of the integrand \(t - 5^t\). Although \(t\) is an algebraic function, \(5^t\) is transcendental because it involves exponential growth. Understanding these kinds of functions is crucial, as they frequently appear in calculus involving integration and differentiation.

Power Rule

The power rule is a fundamental concept in calculus for finding derivatives, but it can also apply to integrals, specifically polynomial integrals. When integrating a function like \(t\), you increase the power by one and divide by the new power.
For instance, the integral \(\int t \, dt\) resolves using the power rule into \(\frac{t^2}{2}\), as seen in Step 2 of the solution. This rule simplifies the process of integrating polynomial functions by providing a straightforward mechanism to follow, avoiding lengthy calculations and making it easy to handle powers.

Special Integral Rule

Special integral rules are necessary for integrating functions that don't fit neatly into the standard polynomial logic, especially for exponential functions with a base other than the natural number \(e\).
In this case, the function \(5^t\) in the exercise uses a special integral rule: \(\int a^t \, dt = \frac{a^t}{\ln(a)}\). This formula is derived from the natural logarithm concept and must be applied when the base \(a\) isn't the natural number \(e\). This special rule helps solve integrals where traditional methods aren't applicable, thus expanding the essential toolbox for solving complex integrals.

Graphing Utility

A graphing utility is a digital tool used to visualize and confirm mathematical solutions, especially integrals. By using such a utility, you can graph the function and compare the observed area with the analytically calculated integral.
This step is crucial because it provides a visual reaffirmation of the mathematically derived result. In the exercise, after calculating the definite integral of \(t - 5^t\) from 0 to 3, the graphing utility helps verify this result by showing the area under the curve. Through this verification, any potential errors in manual calculation can be identified and corrected, making graphing utilities indispensable in modern calculus.