Problem 34 - Calculus Single Variable

Question

Calculate the given definite integral. \(\int_{0}^{1} \frac{2^{x}+3^{x}}{5^{x}} d x\)

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{-\frac{3}{5}}{\ln\left(\frac{2}{5}\right)} + \frac{-\frac{2}{5}}{\ln\left(\frac{3}{5}\right)} \).

Step 1: Simplify the Integrand

The function to integrate is \( \frac{2^x + 3^x}{5^x} \). Rewrite the integrand as a sum of two fractions: \( \frac{2^x}{5^x} + \frac{3^x}{5^x} \). Each fraction can further be simplified to \( \left(\frac{2}{5}\right)^x \) and \( \left(\frac{3}{5}\right)^x \) respectively by using the property \( \frac{a^x}{b^x} = \left(\frac{a}{b}\right)^x \). Thus, the integrand becomes \( \left(\frac{2}{5}\right)^x + \left(\frac{3}{5}\right)^x \).

Step 2: Split the Integral

Now split the integral \( \int_{0}^{1} \left(\frac{2}{5}\right)^x + \left(\frac{3}{5}\right)^x \, dx \) into two separate integrals: \( \int_{0}^{1} \left(\frac{2}{5}\right)^x dx + \int_{0}^{1} \left(\frac{3}{5}\right)^x dx \).

Step 3: Integrate \( \left(\frac{2}{5}\right)^x \)

Use the formula for integrating \( a^x \), which is \( \int a^x \, dx = \frac{a^x}{\ln(a)} + C \). For \( \left(\frac{2}{5}\right)^x \), apply this formula: \( \int \left(\frac{2}{5}\right)^x \, dx = \frac{\left(\frac{2}{5}\right)^x}{\ln\left(\frac{2}{5}\right)} + C \). Evaluate this from 0 to 1.

Step 4: Evaluate \( \int_{0}^{1} \left(\frac{2}{5}\right)^x dx \)

Calculate the definite integral: \[ \left[ \frac{\left(\frac{2}{5}\right)^x}{\ln\left(\frac{2}{5}\right)} \right]_{0}^{1} = \frac{\left(\frac{2}{5}\right)^1}{\ln\left(\frac{2}{5}\right)} - \frac{\left(\frac{2}{5}\right)^0}{\ln\left(\frac{2}{5}\right)} = \frac{\frac{2}{5}}{\ln\left(\frac{2}{5}\right)} - \frac{1}{\ln\left(\frac{2}{5}\right)} \]. Factor out \( \frac{1}{\ln\left(\frac{2}{5}\right)} \): \[ \frac{1}{\ln\left(\frac{2}{5}\right)} \left( \frac{2}{5} - 1 \right) = \frac{-\frac{3}{5}}{\ln\left(\frac{2}{5}\right)} \].

Step 5: Integrate \( \left(\frac{3}{5}\right)^x \)

Similarly, apply the integration formula: \( \int \left(\frac{3}{5}\right)^x \, dx = \frac{\left(\frac{3}{5}\right)^x}{\ln\left(\frac{3}{5}\right)} + C \). Evaluate this from 0 to 1.

Step 6: Evaluate \( \int_{0}^{1} \left(\frac{3}{5}\right)^x dx \)

Compute the definite integral: \[ \left[ \frac{\left(\frac{3}{5}\right)^x}{\ln\left(\frac{3}{5}\right)} \right]_{0}^{1} = \frac{\left(\frac{3}{5}\right)^1}{\ln\left(\frac{3}{5}\right)} - \frac{\left(\frac{3}{5}\right)^0}{\ln\left(\frac{3}{5}\right)} = \frac{\frac{3}{5}}{\ln\left(\frac{3}{5}\right)} - \frac{1}{\ln\left(\frac{3}{5}\right)} \]. Factor out \( \frac{1}{\ln\left(\frac{3}{5}\right)} \): \[ \frac{1}{\ln\left(\frac{3}{5}\right)} \left( \frac{3}{5} - 1 \right) = \frac{-\frac{2}{5}}{\ln\left(\frac{3}{5}\right)} \].

Step 7: Combine the Results

Sum the results from the two integrals to get the final answer: \[ \frac{-\frac{3}{5}}{\ln\left(\frac{2}{5}\right)} + \frac{-\frac{2}{5}}{\ln\left(\frac{3}{5}\right)} \]. This gives the value of the original definite integral.

Key Concepts

Integration Techniques

When solving calculus problems, especially involving definite integrals, choosing the right integration techniques is crucial. This exercise showcases the technique of splitting the integral into simpler, more manageable parts.
We start with the original problem: \[ \int_{0}^{1} \frac{2^{x}+3^{x}}{5^{x}} \ d x \]Rather than tackling the integral head-on, we split the function into two easier components. By transforming the integrand to:\[ \left(\frac{2}{5}\right)^{x} + \left(\frac{3}{5}\right)^{x} \],we essentially reduce complexity.
Each part becomes a standalone exponential function, which is much easier to integrate. This maneuver is particularly useful when dealing with complex expressions.

• Breaking down complex integrals into simpler parts makes integration easier.
• Look out for fractions that can be expressed as simpler exponentials.

Techniques like this highlight how a good strategy can transform an initially challenging problem into a series of straightforward calculations.

Exponential Functions

Understanding exponential functions is key to solving definite integrals like the one presented. In this exercise, we deal with forms like: \[ \left(\frac{2}{5}\right)^{x} \text{ and } \left(\frac{3}{5}\right)^{x} \].
Each of these is an exponential function, characterized by a constant base raised to a variable exponent.Exponential functions have the general form \( a^x \), where \( a \) is a positive real number. When integrating these, we use a specific rule:\[ \int a^{x} \, dx = \frac{a^{x}}{\ln(a)} + C \]This formula arises because the derivative of \( \ln(a^x) \) involves \( \ln(a) \). This is why the natural logarithm appears in the denominator.

• Recognize the form of an exponential to apply the correct integral formula directly.
• Remember, the base of the exponential function must be positive and different from 1.

Consistently applying these principles ensures that exponential functions, despite their complexity, can be integrated smoothly within calculus.

Calculus Problem Solving

Solving calculus problems, like the provided definite integral, hinges on a thorough understanding of various mathematical concepts and techniques. Here, we combined simplification, splitting the integral, and applying exponential integration.
To effectively tackle such problems:

• First, simplify the problem as much as possible. Here, by breaking down the integrand into a sum of exponential functions.
• Second, apply appropriate mathematical properties and formulas. We used the formula for integrating exponential functions.
• Finally, evaluate the solution accurately. This involves computing the difference between the values at the bounds, as shown in each step.

Remember, calculus problem solving is about recognizing patterns, understanding when and how to apply formulas, and methodically working through each component of the problem. By dissecting the integration process, students gain a clearer grasp of calculus, allowing them to solve increasingly complex problems with confidence.