Problem 3 - Calculus Early Transcendentals

Question

Evaluate \(\int 4^{x} d x\).

Step-by-Step Solution

Answer

Question: Evaluate the integral of the function \(f(x) = 4^{x}\). Answer: \(\int 4^{x} dx = \frac{1}{\ln(4)}4^x + C\)

Step 1: Identify the integrand

We are given the integral \(\int 4^{x} dx\). The function we want to integrate is \(f(x) = 4^{x}\).

Step 2: Determine the substitution

In this case, we need to evaluate the integral in terms of exponential functions \(a^x\) with base \(e\). We can use the property that \(a^x = e^{\ln(a^x)} = e^{x\ln(a)}\). So, we can rewrite our integral as follows: \(\int 4^{x} dx = \int e^{x\ln(4)} dx\)

Step 3: Perform the substitution

Let \(u = x\ln(4)\). So, \(\frac{du}{dx} = \ln(4)\). Then, our integral becomes: \(\int e^u \frac{1}{\ln(4)} du\)

Step 4: Integrate

Now, we can integrate the function: \(\int e^u \frac{1}{\ln(4)} du = \frac{1}{\ln(4)}\int e^u du\) The integral of \(e^u\) is simply \(e^u\). So, we have: \(\frac{1}{\ln(4)}\int e^u du = \frac{1}{\ln(4)}e^u + C\) where \(C\) is the constant of integration.

Step 5: Substitute back

Now, we substitute back our original variable \(x\): \(\frac{1}{\ln(4)}e^{x\ln(4)} + C = \frac{1}{\ln(4)}4^x + C\) Therefore, the evaluated integral is: \(\int 4^{x} dx = \frac{1}{\ln(4)}4^x + C\)