# How do you integrate #g(x)=sinx/x^2# using the quotient rule?

See explanation.

From the topic you wrote in I assume you mean "how to differentiate the function?"

For the given function the derivative is:

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You integrate ( g(x) = \frac{\sin(x)}{x^2} ) using integration by parts, not the quotient rule.

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To integrate ( g(x) = \frac{\sin(x)}{x^2} ) using the quotient rule, we first rewrite the integral as:

[ \int \frac{\sin(x)}{x^2} , dx ]

Now, applying the quotient rule, we have:

[ \int \frac{u}{v} , dx = \int u \cdot \frac{1}{v} , dx ]

Where ( u = \sin(x) ) and ( v = x^2 ).

Next, we need to find ( du ) and ( dv ):

[ du = \frac{d(\sin(x))}{dx} = \cos(x) , dx ]

[ dv = \frac{d(x^2)}{dx} = 2x , dx ]

Now, we apply the integration by parts formula:

[ \int u , dv = uv - \int v , du ]

Substituting the values:

[ \int \sin(x) \cdot \frac{1}{x^2} , dx = -\frac{\cos(x)}{x} - \int -\frac{2x \cos(x)}{x^2} , dx ]

Simplify the integral:

[ \int \frac{\sin(x)}{x^2} , dx = -\frac{\cos(x)}{x} + 2 \int \frac{\cos(x)}{x} , dx ]

Now, we have the integral of ( \frac{\cos(x)}{x} ), which does not have a simple closed-form antiderivative and is known as the sine integral function ( \text{Si}(x) ). So, we can rewrite it as:

[ \int \frac{\cos(x)}{x} , dx = \text{Si}(x) + C ]

Where ( C ) is the constant of integration.

Therefore, the final result of integrating ( g(x) = \frac{\sin(x)}{x^2} ) using the quotient rule is:

[ \int \frac{\sin(x)}{x^2} , dx = -\frac{\cos(x)}{x} + 2 \text{Si}(x) + C ]

Where ( C ) is the constant of integration.

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When evaluating a one-sided 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 one-sided 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 one-sided 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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