Problem 33

Question

Evaluate the integrals in Exercises $29-50.$ $$\int 8 e^{(x+1)} d x$$

Step-by-Step Solution

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Answer

The integral is $8 e^{(x+1)} + C$.

1Step 1: Identify the integrand

The integrand in the problem is given as $8 e^{(x+1)}$. This is a function that involves an exponential expression $e^{(x+1)}$, scaled by a constant factor 8.

2Step 2: Recognize the integral form

The integral formula for an exponential function is $\int e^{u} \, du = e^{u} + C$, where $u$ is a function of $x$. Our task is to bring our integrand to this form. Here, $u = x + 1$.

3Step 3: Apply the substitution method for integration

Since $u = x + 1$, compute the derivative: $ \frac{du}{dx} = 1 $. This implies $du = dx$. The integral becomes $\int 8 e^{u} \, du$.

4Step 4: Integrate the simplified expression

Using the rule for integrating exponential functions, $\int 8 e^{u} \, du = 8 e^{u} + C$.

5Step 5: Substitute back to express in terms of x

Substitute $u = x + 1$ back into our integrated expression: $8 e^{u} + C = 8 e^{(x+1)} + C$.

6Step 6: Write the final solution

The evaluated integral is $8 e^{(x+1)} + C$, where $C$ is the constant of integration.

Key Concepts

exponential functionsubstitution methodintegral of exponential function

exponential function

An exponential function is a type of mathematical function involving a constant base raised to a variable exponent. The most common exponential function is denoted as $e^x$, where $e$ is approximately equal to 2.71828. This is known as Euler's number, and it comes up frequently in calculus and complex analysis because it has unique properties that make differentiation and integration straightforward.
For any exponential function of the form $e^{(x+a)}$, where $a$ is a constant, the function can be rewritten using the property of exponents: $e^x \cdot e^a$. This aspect makes them adaptable for various mathematical manipulations.

Exponential functions are continuous and smooth, with a rapid growth or decay.
They are essential in describing real-world phenomena, such as population growth and radioactive decay.
The derivative of $e^x$ is itself, making it unique among functions.

Understanding exponential functions thoroughly is key in learning how to work with integrals involving these functions.

substitution method

The substitution method is a technique used in calculus to simplify integrals and solve them more easily. It involves substituting a part of the integral with a new variable to transform it into a simpler form. In our exercise, we use the substitution $u = x + 1$, making the integral easier to evaluate.
The process can be broken down as follows:

Identify a part of the integrand to substitute. Choose a function $u(x)$ such that its derivative is present elsewhere in the integrand.
Calculate the derivative $\frac{du}{dx}$ and express $dx$ in terms of $du$.
Rewrite the original integral in terms of $u$ and $du$.
Integrate with respect to $u$. Once done, substitute back the original variable.

This method is helpful because it can turn a difficult or seemingly unsolvable integral into a simpler one.

integral of exponential function

The integral of an exponential function like $e^u$ is straightforward because the integral preserves the exponential form. The basic rule is:\[\int e^u \, du = e^u + C\]Here, $C$ is the constant of integration that appears in indefinite integrals. Integrating an exponential function essentially reverses the differentiation, since the derivative of $e^x$ is also $e^x$.
In the provided exercise, after performing substitution, the integral $\int 8 e^u \, du$ results in $8e^u + C$. This shows the speed and ease with which exponential functions can be integrated, given they retain their format.

Exponential integrals maintain their simple form, making them uniquely predictable.
The constant multiplier can be factored out, as seen with the $8$ in the integral $8e^u$.
Understanding integrals like these is foundational for tackling more complex calculus problems involving growth and decay models.

Mastering this concept will make handling exponential functions in integration more intuitive and efficient.

Problem 33

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