CALCU-f(x)= x/sin(x)1. Calculate the first deriva

f(x)= x/sin(x) 1. Calculate the first derivative of the function. 2. Find the indefinite integral of the function.

Solution by Steps

step 1

To calculate the first derivative of

f(x) = \frac{x}{\sin(x)}

, we will use the quotient rule for differentiation, which states that

(f/g)' = \frac{f'g - fg'}{g^2}

, where

f(x) = x

and

g(x) = \sin(x)

step 2

Differentiate

f(x)

with respect to

x

:

f'(x) = \frac{1 \cdot \sin(x) - x \cdot \cos(x)}{\sin^2(x)}

step 3

Simplify the expression:

f'(x) = \frac{\sin(x) - x \cos(x)}{\sin^2(x)}

Answer

f'(x) = \frac{\sin(x) - x \cos(x)}{\sin^2(x)}

Key Concept

Quotient Rule for Differentiation

Explanation

The first derivative of the function

f(x) = \frac{x}{\sin(x)}

is found using the quotient rule, which requires differentiating the numerator and the denominator separately and then combining them according to the rule.

Solution by Steps

step 1

To find the indefinite integral of

f(x) = \frac{x}{\sin(x)}

, we recognize that this is a non-standard integral that does not have an elementary antiderivative

step 2

We can attempt integration by parts or other methods, but they will not yield a simple closed-form expression

step 3

Therefore, the indefinite integral of

f(x) = \frac{x}{\sin(x)}

is not expressible in terms of elementary functions

Answer

The indefinite integral of

f(x) = \frac{x}{\sin(x)}

cannot be expressed in terms of elementary functions.

Key Concept

Non-elementary Antiderivative

Explanation

Some functions do not have antiderivatives that can be expressed in terms of elementary functions, and

f(x) = \frac{x}{\sin(x)}

is one such example.

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What is the first derivative of the function

\frac{x}{\sin(x)}

?

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