Problem 14
Question
Finding antiderivatives Find all the antiderivatives of the following functions. Check your work by taking derivatives. $$g(x)=-4 \cos 4 x$$
Step-by-Step Solution
Verified Answer
Answer: The complete set of antiderivatives for the function \(g(x) = -4\cos(4x)\) are given by the expression \( -\sin(4x) + C\), where \(C\) is a constant.
1Step 1: Recall the antiderivative of cosine function
Recall the basic antiderivative formula for cosine function:
$$\int \cos(ax) dx = \frac{1}{a}\sin(ax) + C$$
where \(C\) represents the constant of integration.
2Step 2: Find the antiderivative of the given function
Apply the antiderivative formula for the cosine function to the given function:
$$\int -4 \cos(4x) dx = -4 \int \cos(4x) dx$$
Using the formula we got in Step 1:
$$-4 \int \cos(4x) dx = -4 \cdot \frac{1}{4}\sin(4x) + C$$
Simplify the expression:
$$-4 \cdot \frac{1}{4}\sin(4x) + C = -\sin(4x) + C$$
The antiderivative of \(g(x)=-4\cos(4x)\) is \(-\sin(4x) + C\).
3Step 3: Check the result
To check our result, take the derivative of the antiderivative we found in Step 2:
$$\frac{d}{dx}(-\sin(4x) + C)$$
Apply the chain rule to find the derivative of the \(-\sin(4x)\) term and remember that the derivative of the constant term, \(C\), is zero:
$$\frac{d}{dx}(-\sin(4x)) = -4\cos(4x)$$
The derivative of the antiderivative is the same as the original function, so our result is correct.
The complete set of antiderivatives for the function \(g(x) = -4\cos(4x)\) are given by the expression:
$$-\sin(4x) + C$$
Key Concepts
Cosine FunctionIntegrationChain Rule
Cosine Function
The cosine function is a fundamental trigonometric function that describes the relationship between an angle and the lengths of the sides in a right-angled triangle. It is defined for all real numbers and its values oscillate between -1 and 1.
This function is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) radians. The general form of the cosine function is given as \(\cos(x)\), where \(x\) is the angle in radians. In transformations, coefficients and phase shifts can change the appearance of \(\cos(x)\) on the graph.
This function is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) radians. The general form of the cosine function is given as \(\cos(x)\), where \(x\) is the angle in radians. In transformations, coefficients and phase shifts can change the appearance of \(\cos(x)\) on the graph.
- The basic properties of the cosine function help in solving various calculus problems including differentiation and integration.
- When solving integrals involving \(\cos(ax)\), it is valuable to remember that integrating cosine typically results in the sine function.
Integration
Integration is the process of finding integrals, which are the reverse operation of differentiation. In simple terms, it is the process of finding antiderivatives, or the original function, from its derivative.
The indefinite integral, or antiderivative, of a function \(f(x)\) is represented by \(\int f(x) \, dx\), and involves finding a family of functions whose derivatives are \(f(x)\). A constant \(C\) is often added to the result, representing the infinite number of functions slightly shifted vertically on the graph.
The indefinite integral, or antiderivative, of a function \(f(x)\) is represented by \(\int f(x) \, dx\), and involves finding a family of functions whose derivatives are \(f(x)\). A constant \(C\) is often added to the result, representing the infinite number of functions slightly shifted vertically on the graph.
- For trigonometric functions like cosine, the process involves specific rules.
- The antiderivative of \(\cos(x)\) is \(\sin(x) + C\).
- When constants multiply a cosine function, they factor into the integration, as seen with \(-4\cos(4x)\).
Chain Rule
The chain rule is a fundamental theorem for computing the derivative of the composition of two or more functions. It is especially useful when dealing with integrals that are products or compositions of functions.
When applying this rule in calculus, consider a function \(y = f(g(x))\). The derivative of \(y\) with respect to \(x\) is found using \(f'(g(x)) \cdot g'(x)\). This allows the derivative of more complex nested functions to be calculated effectively.
When applying this rule in calculus, consider a function \(y = f(g(x))\). The derivative of \(y\) with respect to \(x\) is found using \(f'(g(x)) \cdot g'(x)\). This allows the derivative of more complex nested functions to be calculated effectively.
- In the context of differentiation, the chain rule ensures that even when the function inside is complex, such as \(-\sin(4x)\), the correct derivative \(_{-4\cos(4x)}\) can be deduced.
- This application ensures the original function is retrieved accurately through differentiation, therefore verifying the antiderivative process.
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