Problem 46
Question
Express your answers in terms of \(x\). \(\int \log _{2} x d x\)
Step-by-Step Solution
Verified Answer
\( \int \log_2{x} \, dx = \frac{x \ln{x}}{\ln{2}} - \frac{x}{\ln{2}} + C \)
1Step 1: Change of Base for Logarithm
The given integral is \( \int \log_2{x} \, dx \). To solve this problem, we need to express \( \log_2{x} \) in terms of natural logarithms, because it's easier to integrate. Recall the change of base formula: \( \log_a{b} = \frac{\ln{b}}{\ln{a}} \). Thus, we have \( \log_2{x} = \frac{\ln{x}}{\ln{2}} \). This transforms our integral into \( \int \frac{\ln{x}}{\ln{2}} \, dx \).
2Step 2: Pull Out the Constant
Notice \( \frac{1}{\ln{2}} \) is a constant with respect to \( x \). We can factor constants out of integrals: \( \int \frac{\ln{x}}{\ln{2}} \, dx = \frac{1}{\ln{2}} \int \ln{x} \, dx \). Now we focus on solving \( \int \ln{x} \, dx \).
3Step 3: Integration by Parts
To integrate \( \ln{x} \), use the integration by parts formula \( \int u \, dv = uv - \int v \, du \). Choose \( u = \ln{x} \) and \( dv = dx \). Compute \( du = \frac{1}{x} \, dx \) and \( v = x \). Substitute into the formula: \( \int \ln{x} \, dx = x \ln{x} - \int x \cdot \frac{1}{x} \, dx = x \ln{x} - \int 1 \, dx \). This simplifies to \( x \ln{x} - x + C \), where \( C \) is the integration constant.
4Step 4: Substitute Back into the Original Integral
Place the expression from Step 3 into the integral found in Step 2: \( \frac{1}{\ln{2}} (x \ln{x} - x + C) \). Simplify to: \( \frac{x \ln{x}}{\ln{2}} - \frac{x}{\ln{2}} + \frac{C}{\ln{2}} \). This is the antiderivative of the original problem in terms of \( x \).
Key Concepts
Logarithmic IntegrationChange of Base FormulaIntegration by Parts
Logarithmic Integration
Logarithmic integration can be challenging, but it becomes simpler once you know the tricks. When we see an integral involving a logarithm, like \( \int \log_2{x} \, dx \), we can use resources like the change of base formula to make the integration process easier.
In this situation, changing the base of the logarithm to a natural logarithm simplifies the calculation. Using the formula \( \log_a{b} = \frac{\ln b}{\ln a} \), we can transform our integral into an expression using natural logarithms, which are much more familiar and easier to integrate.
In this situation, changing the base of the logarithm to a natural logarithm simplifies the calculation. Using the formula \( \log_a{b} = \frac{\ln b}{\ln a} \), we can transform our integral into an expression using natural logarithms, which are much more familiar and easier to integrate.
- Use the natural log whenever possible to simplify problems.
- Remember that integration rules often rely on classic functions like \( \ln{x} \), making them easier to handle.
Change of Base Formula
The change of base formula is a crucial tool in integral calculus when dealing with logarithms of different bases. This formula allows you to convert a logarithm of one base into a natural logarithm, making integration straightforward.
The formula is stated as:\[\log_a b = \frac{\ln b}{\ln a}\]This transformation is significant because natural logarithms (\( \ln{x} \)) are more convenient to work with, especially in calculus, as they have well-known derivatives and integrals.
The formula is stated as:\[\log_a b = \frac{\ln b}{\ln a}\]This transformation is significant because natural logarithms (\( \ln{x} \)) are more convenient to work with, especially in calculus, as they have well-known derivatives and integrals.
- Converting to natural logarithms aids in applying integration techniques like integration by parts.
- This step reduces the complexity of many logarithmic expressions.
Integration by Parts
Integration by parts is a valuable technique in calculus used to integrate products of functions. The formula for integration by parts is expressed as:\[\int u \, dv = uv - \int v \, du\]To effectively use this method, carefully choose your \( u \) and \( dv \) based on the LIATE rule, which prioritizes Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential functions.
In your exercise, when finding \( \int \ln{x} \, dx \), this formula is applied by setting \( u = \ln{x} \) and \( dv = dx \). This choice simplifies the process, as you then find: \( du = \frac{1}{x} \, dx \) and \( v = x \).
In your exercise, when finding \( \int \ln{x} \, dx \), this formula is applied by setting \( u = \ln{x} \) and \( dv = dx \). This choice simplifies the process, as you then find: \( du = \frac{1}{x} \, dx \) and \( v = x \).
- Substitute these choices into the formula to simplify the integral.
- Break the integral into more manageable parts and simplify step by step.
Other exercises in this chapter
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