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How do you integrate #y=(sin18x)/(6x)# using the quotient rule?

Answer 1

Cameron Alexander

There is no quotient rule for integration.

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Answer 2

Charles Anctil

To integrate (y = \frac{\sin(18x)}{6x}) using the quotient rule, follow these steps:

Apply the quotient rule: ( \int \frac{f'(x)}{g(x)} ,dx = \frac{f(x)}{g(x)} - \int \frac{f(x)g'(x)}{[g(x)]^2} ,dx ).
Identify (f(x)) and (g(x)): (f(x) = \sin(18x)) (g(x) = 6x)
Find the derivatives of (f(x)) and (g(x)): (f'(x) = 18\cos(18x)) (g'(x) = 6)
Substitute into the quotient rule formula: ( \int \frac{18\cos(18x)}{6x} ,dx = \frac{\sin(18x)}{6x} - \int \frac{\sin(18x) \cdot 6}{(6x)^2} ,dx ).
Simplify: ( \int \frac{18\cos(18x)}{6x} ,dx = \frac{\sin(18x)}{6x} - \int \frac{\sin(18x)}{x^2} ,dx ).
Now, integrate ( \int \frac{\sin(18x)}{x^2} ,dx ) using a suitable method, such as substitution.

That's the integration of (y = \frac{\sin(18x)}{6x}) using the quotient rule.

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Answer from HIX Tutor

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.