How do you integrate #f(x)=sinx/x# using the quotient rule?
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To integrate ( f(x) = \frac{\sin(x)}{x} ) using the quotient rule, first differentiate the function ( \frac{\sin(x)}{x} ) with respect to ( x ) to get the derivative, then integrate the result.

Differentiate ( \frac{\sin(x)}{x} ) using the quotient rule: [ \frac{d}{dx}\left(\frac{\sin(x)}{x}\right) = \frac{x \cdot \cos(x)  \sin(x)}{x^2} ]

Now integrate the result: [ \int \frac{x \cdot \cos(x)  \sin(x)}{x^2} , dx ]
This integral can be solved by separating it into two integrals:
[ \int \frac{x \cdot \cos(x)}{x^2} , dx  \int \frac{\sin(x)}{x^2} , dx ]
The first integral can be simplified by canceling out ( x ) from numerator and denominator:
[ \int \frac{\cos(x)}{x} , dx ]
This integral can be solved using techniques such as integration by parts or specialized methods for handling integrals of the form ( \int \frac{\cos(x)}{x} , dx ).
The second integral, ( \int \frac{\sin(x)}{x^2} , dx ), is known as the sine integral and does not have a simple elementary antiderivative. It can be expressed in terms of special functions like the Si(x) function.
Therefore, integrating ( f(x) = \frac{\sin(x)}{x} ) using the quotient rule leads to a more complex expression, involving integrals that may not have simple closedform solutions.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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