Problem 48 - Calculus Single Variable

Question

Find a function whose derivative is the given function. \(\cos (x)\left(5 \csc ^{2}(x)-1\right)\)

Step-by-Step Solution

Answer

The function is \( f(x) = -5 \csc(x) - \sin(x) + C \).

Step 1: Identify the Given Function

The function given is \( f'(x) = \cos(x)(5 \csc^2(x) - 1) \). We need to find a function \( f(x) \) such that \( f'(x) = \cos(x)(5 \csc^2(x) - 1) \).

Step 2: Analyze the Expression

The derivative \( f'(x) = \cos(x)(5 \csc^2(x) - 1) \) can be expressed as a product of \( \cos(x) \) and \( 5 \csc^2(x) - 1 \). This suggests the use of trigonometric identities and integration techniques to find the original function.

Step 3: Simplify and Separate Terms

Re-express the function as two separate terms: \( f'(x) = 5 \cos(x) \csc^2(x) - \cos(x) \). This allows us to integrate each part individually.

Step 4: Integrate First Term

To integrate \( 5 \cos(x) \csc^2(x) \), recognize that \( \csc(x) = 1/\sin(x) \). Therefore, \( \csc^2(x) = 1/\sin^2(x) \). Notice that \( \cos(x)\csc^2(x) = \cot(x) \cdot \csc(x) \), which suggests that the integral is related to \( \csc(x) \), known to be \( -\csc(x) \). Compute the integral: \[ \int 5 \cos(x) \csc^2(x) \, dx = -5 \csc(x) + C_1 \].

Step 5: Integrate Second Term

Now integrate the second term \( -\cos(x) \), which is a standard integral: \[ \int -\cos(x) \, dx = -\sin(x) + C_2 \].

Step 6: Combine Integrals

Combine the results from both integrals: \[ f(x) = -5 \csc(x) - \sin(x) + C \], where \( C = C_1 + C_2 \) is the constant of integration.

Key Concepts

Integration Techniques

Integration techniques are essential tools in calculus for finding antiderivatives, which are the functions whose derivatives match a given function. These techniques help us reverse the process of differentiation. There are several methods available, each suited to different kinds of functions.
One commonly used technique is recognizing standard forms of derivatives. In many cases, experienced mathematicians can identify these forms and know the corresponding antiderivatives by heart.
Another important technique is integration by substitution, sometimes called "u-substitution." This is useful when you can change variables to transform a difficult integral into a simpler one.
Additionally, integration by parts is a valuable method that uses the product rule of derivatives in reverse to simplify integrals involving products.

• Standard forms recognition
• u-substitution
• Integration by parts

In the original exercise, these techniques allow us to separate the given derivative into parts that we can easily manage. By recognizing these types in parts of the function, we find their respective antiderivatives.

Trigonometric Identities

Trigonometric identities play a crucial role in integrating functions that involve trigonometric expressions. They help us simplify or reformat the function, making the integration process easier.
Some fundamental trigonometric identities include:

• Pythagorean identities, such as \(\sin^2(x) + \cos^2(x) = 1\)
• Reciprocal identities, like \(\csc(x) = 1/\sin(x)\)
• Quotient identities, such as \(\tan(x) = \sin(x)/\cos(x)\)

Using these identities, we simplified the original function \(\cos(x)(5\csc^2(x) - 1)\) into manageable parts. For instance, recognizing \(\csc(x) = 1/\sin(x)\) helps us understand and transform \(\csc^2(x)\) into \(1/\sin^2(x)\). This step is critical in transforming the derivative into antiderivative functions that are already known, such as \(-\csc(x)\) and \(-\sin(x)\).

Function Integration

Function integration is the process of calculating the integral, or antiderivative, of a function. This foundational aspect of integral calculus involves reversing differentiation to find a function whose derivative is given.
In this process, constants may appear, which are important to ensure the general solution encapsulates all possible functions that could differentiate into the given one. This constant is often denoted by \( C \).
The procedure in our problem involved splitting the function into simpler parts for easier integration. Breaking down the given function \(\cos(x)(5\csc^2(x) - 1)\) into smaller components allows for the integration of each part separately: \(5\cos(x)\csc^2(x)\) and \(-\cos(x)\).
Once these smaller parts are integrated independently, we add them to construct the full antiderivative function. Importantly, the constant \( C \) is added to represent any constant value that could also differentiate into zero, thus maintaining the integrity of the general solution. Therefore, we achieve the final result: \(-5\csc(x) - \sin(x) + C\).