Problem 53
Question
Find the indefinite integral. $$ \int 3^{x / 2} d x $$
Step-by-Step Solution
Verified Answer
The indefinite integral of \(\int 3^{x/2} dx\) is \(3^{x/2} + C\)
1Step 1: Conversion to Base e
Convert the base of the exponential function to \(e\). This is done using the formula \(a^x = e^{x\ln a}\), where \(a\) is a constant. Hence, we convert \(3^{x/2}\) as \(e^{(x/2)\ln 3}\)
2Step 2: Apply Exponential Integral Rule
The integral of any function in the form \(e^u du\) is always equal to \(e^u + C\), where \(C\) is the constant of integration. Now, our integral looks like \(\int e^{u} du\), where \(u = (x/2)\ln 3\). So, the direct application of the exponential integral rule gives us the result as \(e^{(x/2)\ln 3} + C\).
3Step 3: Convert Back to Original Base
Now, convert the result back to the base 3. So, the result is \(3^{x/2} + C\), where \(C\) is the constant of integration.
Key Concepts
Exponential FunctionIntegration RulesNatural Logarithm
Exponential Function
The exponential function is a mathematical expression in which a constant base is raised to a variable exponent. In the context of integration, exponential functions often appear in the form of the natural exponential function, expressed as \(e^x\), where \(e\), the base, is the natural logarithm's base, approximated as 2.71828.
These functions model growth or decay processes, like population growth or radioactive decay. They have a unique property: the function's derivative is proportional to the function itself. This property makes exponential functions relatively straightforward to integrate compared to other types of functions.
For instance, integrating the function \(3^{x / 2}\), requires converting it into a function of the natural base \(e\), which is achieved through the identity \(a^x = e^{x\ln(a)}\). This conversion simplifies the integration process, which uses the rules of integration that specifically cater to exponential functions with the base \(e\).
These functions model growth or decay processes, like population growth or radioactive decay. They have a unique property: the function's derivative is proportional to the function itself. This property makes exponential functions relatively straightforward to integrate compared to other types of functions.
For instance, integrating the function \(3^{x / 2}\), requires converting it into a function of the natural base \(e\), which is achieved through the identity \(a^x = e^{x\ln(a)}\). This conversion simplifies the integration process, which uses the rules of integration that specifically cater to exponential functions with the base \(e\).
Integration Rules
Integration is a fundamental concept in calculus, serving as the inverse process to differentiation. There are numerous rules in integration that make solving integrals more manageable. Some of these rules include the power rule, the substitution rule, and integration by parts.
For exponential functions, the integral \(\int e^{u} du\) is particularly important. When dealing with functions of the form \(e^u\), where \(u\) is a function of \(x\), one can directly apply the rule that the integral of \(e^u\) is \(e^u + C\), with \(C\) representing the constant of integration.
To apply these rules correctly, attention must be paid to the specific structure of the function being integrated. In the case of \(\int 3^{x / 2} dx\), adopting a strategy that involves rewriting the function in terms of \(e\) makes the problem more tractable. In such contexts, the chain rule for integration often comes into play, requiring the integral of a composite function to be approached through substitution, thereby simplifying the integration process.
For exponential functions, the integral \(\int e^{u} du\) is particularly important. When dealing with functions of the form \(e^u\), where \(u\) is a function of \(x\), one can directly apply the rule that the integral of \(e^u\) is \(e^u + C\), with \(C\) representing the constant of integration.
To apply these rules correctly, attention must be paid to the specific structure of the function being integrated. In the case of \(\int 3^{x / 2} dx\), adopting a strategy that involves rewriting the function in terms of \(e\) makes the problem more tractable. In such contexts, the chain rule for integration often comes into play, requiring the integral of a composite function to be approached through substitution, thereby simplifying the integration process.
Natural Logarithm
The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), which is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm of a number is the exponent to which \(e\) must be raised to produce that number. For instance, \(\ln(e^x) = x\), and conversely, \(e^{\ln(x)} = x\).
The natural logarithm has a direct relationship with exponential functions, especially when integrating expressions that involve other bases, such as in the exercise \(\int 3^{x / 2} dx\). Here, the conversion formula \(a^x = e^{x\ln(a)}\) leverages the natural logarithm to transform the base into \(e\), making the integral solvable with the exponential integral rule.
Understanding the natural logarithm is essential for integration, as it often provides the tool required for solving integrals involving exponential functions, particularly when the function's base is not \(e\). The logarithmic properties such as \(\ln(xy) = \ln(x) + \ln(y)\) and \(\ln(x^y) = y\ln(x)\) are powerful tools in simplifying expressions prior to integration.
The natural logarithm has a direct relationship with exponential functions, especially when integrating expressions that involve other bases, such as in the exercise \(\int 3^{x / 2} dx\). Here, the conversion formula \(a^x = e^{x\ln(a)}\) leverages the natural logarithm to transform the base into \(e\), making the integral solvable with the exponential integral rule.
Understanding the natural logarithm is essential for integration, as it often provides the tool required for solving integrals involving exponential functions, particularly when the function's base is not \(e\). The logarithmic properties such as \(\ln(xy) = \ln(x) + \ln(y)\) and \(\ln(x^y) = y\ln(x)\) are powerful tools in simplifying expressions prior to integration.
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