Problem 74
Question
$$\frac{-\sin ^{-1} a x}{x}-a \ln \left|\frac{1+\sqrt{1-(a x)^{2}}}{a x}\right|+C$$
Step-by-Step Solution
Verified Answer
Answer: The main components of the given function are:
1. $$-\frac{\sin^{-1}(ax)}{x}$$: This component represents the inverse sine function of the product "ax" divided by the variable x.
2. $$-a\ln\left|\frac{1+\sqrt{1-(ax)^2}}{ax}\right|$$: This component comprises a natural logarithm function with a term inside the absolute value, multiplied by the variable "a".
3. $$C$$: This is an unspecified constant term, added because of possible integration.
These components may be connected through the process of integration, although without a specific context, further analysis or application of the function is limited.
1Step 1: 1. Identify the components of the function
The given function consists of three main components:
1. $$-\frac{\sin^{-1}(ax)}{x}$$
2. $$-a\ln\left|\frac{1+\sqrt{1-(ax)^2}}{ax}\right|$$
3. $$C$$
2Step 2: 2. Understand the first component: $$-\frac{\sin^{-1}(ax)}{x}$$
This component consists of the inverse sine function and the variable x in the denominator. The inverse sine function, which is written as $$\sin^{-1}(ax)$$ or arcsin(ax), refers to the angle whose sine is equal to $$ax$$. Also, this function is being multiplied by $$-1$$ in the given expression.
3Step 3: 3. Understand the second component: $$-a\ln\left|\frac{1+\sqrt{1-(ax)^2}}{ax}\right|$$
This component consists of:
1. A variable "a" multiplied by the entire expression.
2. A natural logarithm function, represented by "ln" with a term in absolute value | | inside the logarithm.
3. A fraction where the numerator is a sum composed of "1" and a square root term ($$1 - (ax)^2$$) and the denominator is "ax".
The exponential function of the natural logarithm is described as the natural base "e" raised to the power of the exponent inside the logarithm. It's important to note that the absolute value is included to ensure that the logarithm's input is always positive.
4Step 4: 4. Understand the third component: Constant C
The last term "C" in the given expression is a constant. It is added because when integrating a function, there can be an arbitrary constant that doesn't affect the derivative (in case we want to find the derivative of the given function).
5Step 5: 5. Connection between the components
The given expression might be the result of integrating a function, since we have the sum of two different functions and an added constant C. Therefore, it may be possible to connect the components by differentiating the given expression with respect to the variable x.
Now that we have broken down and understood the individual components of the given function, it's important to remember that without a specific context or problem statement, further analysis or application is limited.
Key Concepts
Inverse Trigonometric FunctionsNatural LogarithmIntegration
Inverse Trigonometric Functions
Inverse trigonometric functions, such as the inverse sine (written as \(\sin^{-1}(x)\) or \(\text{arcsin}(x)\)), help find an angle when given a trigonometric ratio. For \(\sin^{-1}(ax)\), you're determining the angle whose sine is \(ax\). This function is often encountered when working with integration problems involving trigonometric functions. It's crucial in calculus because it allows us to handle scenarios where reversing the trigonometric process is necessary.
Using inverse trigonometric functions can be a bit tricky due to their range restrictions. For example, \(\sin^{-1}(x)\) only outputs angles within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This concept is especially useful in solving integrals where trigonometric identities can greatly simplify the process and make integration more manageable.
Using inverse trigonometric functions can be a bit tricky due to their range restrictions. For example, \(\sin^{-1}(x)\) only outputs angles within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This concept is especially useful in solving integrals where trigonometric identities can greatly simplify the process and make integration more manageable.
Natural Logarithm
The natural logarithm, denoted as \(\ln\), is a logarithm with the base \(e\) (approximately 2.71828). It's a fundamental tool in calculus, and it comes up frequently in integration due to its special properties. The natural log of a number is essentially the power to which \(e\) must be raised to get that number.
The expression \(a\ln\left|\frac{1+\sqrt{1-(ax)^2}}{ax}\right|\) involves a natural logarithm, ensuring that the input of the \(\ln\) function is positive through the absolute value signs \(|\cdots|\). This ensures valid operations on real numbers. Using logs in integration can simplify expressions, making them easier to integrate, especially when dealing with products or quotients inside the logarithm.
The expression \(a\ln\left|\frac{1+\sqrt{1-(ax)^2}}{ax}\right|\) involves a natural logarithm, ensuring that the input of the \(\ln\) function is positive through the absolute value signs \(|\cdots|\). This ensures valid operations on real numbers. Using logs in integration can simplify expressions, making them easier to integrate, especially when dealing with products or quotients inside the logarithm.
Integration
Integration is one of the core operations in calculus, closely related to differentiation. It involves finding the integral or antiderivative of a function, which can be visualized as the area under a curve.
In this context, the compound expression given might be derived from integrating a related function. Integration can be challenging but is made simpler with techniques such as substitution, integration by parts, and recognizing inverse trigonometric or logarithmic functions that form specific patterns. The constant \(C\) in the integrated expression is critical because it represents an infinite number of possible constants that can be added to an indefinite integral. This constant ensures that the integration process covers all possible solutions, keeping in mind that antiderivatives are not unique without initial conditions.
In this context, the compound expression given might be derived from integrating a related function. Integration can be challenging but is made simpler with techniques such as substitution, integration by parts, and recognizing inverse trigonometric or logarithmic functions that form specific patterns. The constant \(C\) in the integrated expression is critical because it represents an infinite number of possible constants that can be added to an indefinite integral. This constant ensures that the integration process covers all possible solutions, keeping in mind that antiderivatives are not unique without initial conditions.
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