Question

Use the Exponential Rule to find the indefinite integral. $$ \int 3 e^{-(x+1)} d x $$

Short answer

\[ -3e^{-(x+1)} + C \]

Step-by-step solution

Identify a suitable substitution

Firstly, in order to simplify the integral, a substitution can be made. One common approach is to let \( u = -(x + 1) \), then compute the differential \( du \), which would be \( -dx \).

Rewrite the integral in terms of 'u'

Next, replace \( -(x + 1) \) with \( u \) and change \( dx \) in terms of \( du \). Thus the integral now becomes \( -\int 3e^u du \). Note the negative sign from the \( du \) calculation is transferred to the front of the integral.

Apply the Exponential Rule

The Exponential Rule states that the integral of \( e^u du \) is \( e^u + C \) where \( C \) is the constant of integration. Apply this rule to the integral to get \( -3e^u + C \).

Substitute back for 'x'

Final step is to replace all instances of \( u \) with \( -(x + 1) \) to get the solution in terms of the original variable \( x \). The final answer becomes \( -3e^{-(x+1)} + C \).

Key concepts

Exponential Rule

Understanding the Exponential Rule is key when working with integrals involving exponential functions. In calculus, when we come across an integral of the form \( \int e^u du \), we can apply this rule which simply states that the integral of the exponential function \( e^u \) with respect to \( u \) is itself, \( e^u \), plus the constant of integration, \( C \). This directly results from the fact that the derivative of \( e^u \) is also \( e^u \), making the process of integration the reverse of differentiation in this case.

In the exercise given, we utilize this rule after we've accounted for a substitution that simplifies our original integral of \( 3e^{-(x+1)} \) into a form that directly applies this rule, allowing for a straightforward integration process.

Integration by Substitution

When faced with a complex integral, integration by substitution, also known as \( u \) substitution, is a technique oftentimes employed to simplify the integral into a more manageable form. The method involves identifying a part of the integrand that can be replaced with a variable, typically \( u \) , and expressing the differential \( dx \) in terms of \( du \).

The success of this technique relies on the choice of \( u \) which should simplify the integral when replaced. In our exercise, the substitution \( u = -(x + 1) \) turns the integral into one that is easier to handle, especially with the Exponential Rule in mind. After integrating with respect to \( u \), we should always remember to substitute back the original variables to obtain the final answer in terms of \( x \).

Notably, it’s important to adjust the integral’s limits if given a definite integral with numeric bounds. However, when dealing with an indefinite integral as in this exercise, we focus on finding the most general antiderivative, including the constant of integration in our final expression.

Constant of Integration

The constant of integration, denoted as \( C \), is a fundamental aspect of indefinite integrals. Since the derivative of any constant is zero, there are infinitely many antiderivatives for any given function, differing only by a constant. Hence, when we find an indefinite integral, it is essential to add \( C \) to represent these possible antiderivatives.

The presence of \( C \) accounts for the fact that when differentiating, information about vertical shifts of the original function is lost. By integrating, we are essentially reversing the process of differentiation, and the constant serves as a placeholder for all potential starting points on the \( y \) axis. In practice, when a specific initial condition is provided, we can find the exact value of \( C \). But in the absence of initial conditions, as shown in the exercise, we include the constant \( C \) to indicate the general solution to the indefinite integral.