Problem 1 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

$$ \int t e^{1-t} d t $$

Step-by-Step Solution

Answer

-e^{1-t}(t + 1) + C

Step 1: Identify the technique

The given integral \( \int t e^{1-t} \, dt \) requires the use of integration by parts. Recall the formula: \[ \int u \, dv = uv - \int v \, du \]

Step 2: Choose \(u\) and \(dv\)

Choose \( u = t \) and \( dv = e^{1-t} \, dt \), so that we can use the integration by parts formula.

Step 3: Compute \(du\) and \(v\)

Differentiate \(u\text{ to get }du \): \[ du = dt \]Now integrate \( dv \text{ to find } v \): \[ v = \int e^{1-t} dt \]

Step 4: Integrate \(dv\)

Perform the integral \ \ \[ v = -e^{1-t} \]

Step 5: Apply the integration by parts formula

Substitute the parts into the integration by parts formula: \[ \int t e^{1-t} dt = t (-e^{1-t}) - \int (-e^{1-t}) \, dt \]Simplify: \[ -t e^{1-t} + \int e^{1-t} dt \]

Step 6: Integrate the remaining term

Integrate the remaining term: \[ \int e^{1-t} dt = -e^{1-t} \]

Step 7: Combine the results

Put all parts together: \[ -t e^{1-t} - e^{1-t} + C \]Combine like terms: \[ -e^{1-t}(t + 1) + C \]

Key Concepts

Definite Integral

In calculus, a definite integral represents the area under the curve of a function from one point to another. It's expressed as \(\textstyle \begin{array}{c} \end{array}\textstyle\bg{[} \int_{a}^{b} f(x) \, dx \bg{]}\textstyle\). This integral provides a numerical value which calculates the net area between the x-axis and the curve of the function within the specified limits.Definite integrals are essential for various applications:

• Finding area under a curve
• Calculating displacement in physics
• Computing accumulated quantities

To compute definite integrals, ensure you:

• Determine the limits of integration \(a \text{ and } b\)
• Use the Fundamental Theorem of Calculus
• Evaluate the antiderivative at these limits

Remember, the definite integral accounts for the area below the x-axis as negative, which can influence the total area calculation.

Indefinite Integral

An indefinite integral, or antiderivative, represents a family of functions and is expressed as \( \int f(x) \, dx \). Instead of computing the area under a curve like in definite integrals, indefinite integrals focus on finding the general form of a function whose derivative is equal to the original function.The result includes a constant of integration, \(C\), because differentiation of a constant is zero:

• Example: The antiderivative of \(f'(x) = 3x^2 \) is \(F(x) = x^3 + C \)

The process of finding an indefinite integral generally involves:

• Recognizing the type of integral (e.g., polynomial, trigonometric)
• Applying integration rules and techniques

Common techniques include:

• Basic integration
• Substitution
• Integration by parts

In our exercise, we used integration by parts, a method particularly useful when handling the product of two functions.

Advanced Calculus

Advanced calculus delves into concepts that build on foundational calculus, such as integration techniques, multivariable calculus, and differential equations. Integration by parts, used in our exercise, is an essential technique in advanced calculus.Integration by parts relies on the formula:\textyle\(\int u \, dv = uv - \int v \, du\)

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

Compute \(du\) and integrate \(dv\) to get \(v\)

Apply the formula

In our exercise:

• \textyle t \text{was chosen as} u\textyle
• \textyle e^{1-t} \, dt \text{was chosen as} dv\textyle
• \textyle v \text{is found to be} -e^{1-t}\textyle

This strategy simplified our integral. These advanced techniques are crucial for tackling complex real-world problems in physics, engineering, and beyond.