Question

In the following exercises, compute each indefinite integral. $$ \int 3^{-x} d x $$

Short answer

The indefinite integral is \(-\frac{3^{-x}}{\ln 3} + C\).

Step-by-step solution

Identify the Integral Form

We need to compute the indefinite integral \( \int 3^{-x} \, dx \). This is an exponential function, which can be integrated using the formula for the integral of an exponential function.

Recognize the Formula

For an exponential function in the form \( a^{bx} \), the integral is \( \frac{a^{bx}}{b \ln a} + C \), where \( C \) is the constant of integration.

Apply the Formula

Here, \( a = 3 \) and \( b = -1 \), so substitute these values into the formula: \[\int 3^{-x} \, dx = \frac{3^{-x}}{-1 \ln 3} + C = -\frac{3^{-x}}{\ln 3} + C.\]

Simplify the Expression

The expression \(-\frac{3^{-x}}{\ln 3} + C \) is already simplified. Therefore, the integral of \( 3^{-x} \) with respect to \( x \) is: \[-\frac{3^{-x}}{\ln 3} + C.\]

Key concepts

Exponential Functions

Exponential functions are a crucial type of function commonly encountered in mathematics. These functions have a distinct form where a constant base, let's call it \(a\), is raised to a variable exponent, typically expressed as \(x\). In general, an exponential function is written as \(a^x\). This differs from linear or polynomial functions, which involve adding powers of \(x\) instead of exponentiating with \(x\).Key characteristics of exponential functions include:

• The base \(a\) is a constant, and in many cases, it is greater than zero. Negative bases are less common and typically lead to complex results.
• Exponential functions grow very quickly. As \(x\) increases, the values of \(a^x\) will also increase rapidly if \(a > 1\).
• If the base is between 0 and 1, the function instead decreases, rapidly approaching zero as \(x\) becomes large.

The exponential function used in our exercise is \(3^{-x}\). This specific case highlights how negative exponents signify an inverse relationship and cause the function to decay as \(x\) grows.

Integration Techniques

Integrating exponential functions can sometimes be tricky, but there is often a direct approach, especially when dealing with bases raised to linear expressions. The formula used in integrating exponential functions revolves around recognizing the form \(a^{bx}\), where \(a\) is the base, \(b\) is the coefficient of \(x\), and the integral is given by:\[\int a^{bx} \, dx = \frac{a^{bx}}{b \ln a} + C\]Here’s how it works:

• Identify the base \(a\) and the linear coefficient \(b\) from your exponential function.
• Substitute these values into the formula above to compute the integral.
• The formula accounts for the chain rule by incorporating a correction factor of \(\frac{1}{b \ln a}\).

By recognizing these steps, you can efficiently handle the integration of exponential functions, even if they first seem complex. In our problem, substituting \(a = 3\) and \(b = -1\) we get the result as:\[-\frac{3^{-x}}{\ln 3} + C\]

Constant of Integration

In calculus, when computing indefinite integrals, the result includes a term known as the "constant of integration," often denoted by \(C\). This "+ C", added to the end of our integral expressions, represents any constant value that makes our antiderivative valid. Here’s why it's important:

• Indefinite integrals are the antiderivatives of a function, and any function can have infinitely many antiderivatives differing by a constant.
• The constant of integration accounts for these possibilities and ensures your result is as general as possible.
• It is essential for calculating definite integrals or applying initial conditions.

In brief, never forget to add \(C\) when solving indefinite integrals like our example \(-\frac{3^{-x}}{\ln 3} + C\), as it completes the expression by acknowledging these infinite antiderivative possibilities.