Problem 6 - Applied Calculus

Question

Find the integrals in problems. Check your answers by differentiation. $$ \int 5 e^{5 t+2} d t $$

Step-by-Step Solution

Answer

The integral is \( e^{5t+2} + C \).

Step 1: Identify the Integral Form

We have the integral \( \int 5 e^{5t+2} \; dt \). This integral involves an exponential function \( e^{5t+2} \), which suggests using the formula for exponential integrals. The general form is \( \int e^{u} \, du = e^{u} + C \), where \( C \) is the integration constant.

Step 2: Apply Substitution Method

Set \( u = 5t + 2 \). Then, differentiate \( u \) with respect to \( t \) to find \( du/dt = 5 \), which gives \( dt = du/5 \). Substitute these into the integral: \( \int 5 e^{u} \cdot \frac{du}{5} \).

Step 3: Simplify and Integrate

The \( 5 \) and \( \frac{1}{5} \) cancel out, so we have \( \int e^{u} \, du \). Integrating this gives \( e^{u} + C \).

Step 4: Substitute Back

Replace \( u \) with the original substitution, \( 5t + 2 \), giving us the final integrated form: \( e^{5t+2} + C \).

Step 5: Differentiate to Check

To verify, differentiate \( e^{5t+2} + C \) with respect to \( t \). By chain rule, \( \frac{d}{dt}[e^{5t+2}] = e^{5t+2} \cdot 5 = 5e^{5t+2} \), which matches the original integrand.

Key Concepts

Exponential Functions

Exponential functions are special because they grow rapidly. In mathematics, an exponential function is a function of the form \( f(x) = a \cdot e^{bx} \), where \( e \) is the base of natural logarithms, approximately equal to 2.71828. This base is special because its derivative is the same as the original function, making it unique in calculus.
When dealing with integrals or derivatives involving exponential functions, the expression typically involves \( e \) raised to some function of \( t \), as seen in the given problem \( 5 e^{5t+2} \). This characteristic makes exponential functions very predictable and pragmatic for solving differential equations. They are extensively used in modeling real-world phenomena involving continuous growth or decay, such as population dynamics, radioactive decay, and compound interest calculations.
Some helpful tips when dealing with exponential functions include:

• Recognize the pattern of the function to apply appropriate calculus techniques.
• Remember that the derivative of an exponential function retains its form, which is useful in checking integration work.

Differentiation

Differentiation is a key concept in calculus that refers to the process of finding the derivative of a function. The derivative measures how a function changes as its input changes, essentially giving us the slope or rate of change of the function.
In our given task, differentiation comes into play when checking the correctness of our integration. Once the integral is found and simplified, we differentiate the resulting expression to ensure it matches the original integrand. This step acts as a validation of our integration process.
Key points to remember about differentiation include:

• Finding the derivative of a function involves identifying and applying different rules, such as the product rule, quotient rule, or chain rule.
• In this example, the chain rule is particularly important due to the compound exponent function \( 5t + 2 \).
• Checking integration with differentiation ensures accuracy, confirming that the integrated function's derivative returns the original function.

Chain Rule

The chain rule is a fundamental differentiation rule used when differentiating compositions of functions. It states that if a function \( y \) depends on \( u \), and \( u \) depends on \( x \), then \( y \) is indirectly dependent on \( x \). The chain rule gives us a way to find how \( y \) changes with \( x \).
In mathematical terms, \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \). In our problem, the use of the chain rule is essential when differentiating \( e^{5t+2} \).
Here’s why the chain rule is applied:

• The exponent \( 5t + 2 \) is itself a function of \( t \), necessitating the use of the chain rule.
• We first differentiate the outer function \( e^u \) (giving \( e^u \)), then multiply by the derivative of the inner function \( 5t + 2 \), which is 5.

This step ensures we capture the effect of the inner function on the rate of change of the outer function.