Question

Find the indefinite integral. $$ \int \frac{5-e^{x}}{e^{2 x}} d x $$

Short answer

The indefinite integral of \(\frac{5-e^{x}}{e^{2 x}} d x\) is \( -5/2e^{-2x}\) - \(e^{-x} + C \)

Step-by-step solution

Separate the terms

Simultaneously simplify the integral by breaking it into two separate terms. The integral becomes \( \int \frac{5}{e^{2x}}dx \) - \( \int \frac{e^{x}}{e^{2x}}dx \)

Further simplify the integrals

The first integral can be simplified to \(5\int e^{-2x}dx \) using the property of the exponent. The second integral simplifies to \(\int e^{-x} dx \)

Substituting variables

Perform a variable substitution in each integral. For the first integral, let \(u=-2x \). For the second integral, let \(v=-x\). After substituting, the integral becomes \( \int 5e^{u}du\) - \( \int e^{v}dv \)

Evaluate the integrals

The integral of \( e^{u} \) with respect to \( u \) is \( e^{u} \) itself. So, we get \( -5/2e^{u} \) from the first integral and \(-e^{v}\) from the second integral.

Substitute back the original variables

Finally, replace the substituted variables \(u\) and \(v\) back with the original variables. The final answer becomes \( -5/2e^{-2x}\) - \(e^{-x} + C \) where \(C\) is the constant of integration.

Key concepts

Integration by Substitution

Integration by substitution, often referred to as u-substitution, is a technique used to solve more complex integrals. It works similarly to the reverse process of the chain rule in differentiation. To perform integration by substitution, we identify a portion of the integrand (the function being integrated) that can be substituted with a new variable, usually denoted as u. This substitution simplifies the integral and makes it easier to evaluate.

Here's the general approach:

• Choose a substitution that simplifies the integral.
• Replace all instances of the original variable with this new variable.
• Change the differential accordingly, so dx becomes du.
• Integrate the simpler function in terms of u.
• Finally, substitute back to the original variable.

Going through the steps helps in transforming a seemingly difficult integral into a more manageable one.

Exponential Functions

Exponential functions, written as e^x, where e is the base of the natural logarithm and x is the exponent, have unique properties that are crucial in calculus, particularly in integration. These functions are their own derivatives, which implies that their antiderivatives are also e^x (up to a constant).

Key aspects of exponential functions include:

• They are always positive, e^x > 0, for all real numbers x.
• They grow rapidly and are used to model growth processes like population and compound interest.
• The integral of e^ax, where a is a constant, is (1/a)e^ax + C, with C being the constant of integration.

Understanding these properties helps in the integration process of functions involving exponents, similar to the exercise provided.

Antiderivatives

An antiderivative of a function is another function whose derivative is the original function. When we integrate a function, we are essentially finding its antiderivative. For example, if F(x) is the antiderivative of f(x), this means that F'(x) = f(x).

Here are some key points:

• The process of finding antiderivatives is called indefinite integration.
• Because of the + C factor, antiderivatives are not unique; they form a family of functions.
• Common antiderivatives should be memorized, such as the antiderivative of x^n, which is (x^(n+1))/(n+1), assuming n ≠ -1.

It's crucial to learn these bases as many integrals will require recognizing and applying the antiderivative of simpler functions as part of the solution process.

Constant of Integration

The constant of integration represents an indefinite number of antiderivatives that a function can have. It's denoted by the symbol C, and it reflects the fact that there are infinitely many functions, all differing by a constant, which have the same derivative.

Here's why it's crucial:

• Whenever we find an antiderivative, we must add + C to indicate that there are multiple solutions.
• The constant of integration is determined only when an initial condition or boundary condition is given, narrowing it down to one specific function.

In the context of our exercise, after finding the indefinite integral of the given function, we add the constant C to encompass all possible antiderivatives of the function.