# How do you integrate #int ln(sint)cost# by integration by parts method?

Now we should apply integration by parts. Let:

Thus:

Integrating and factoring:

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To integrate ( \int \ln(\sin(t)) \cos(t) , dt ) using integration by parts, let ( u = \ln(\sin(t)) ) and ( dv = \cos(t) , dt ). Then, ( du = \frac{\cos(t)}{\sin(t)} , dt ) and ( v = \sin(t) ). Applying the integration by parts formula:

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

we have:

[ \int \ln(\sin(t)) \cos(t) , dt = \ln(\sin(t)) \sin(t) - \int \sin(t) \frac{\cos(t)}{\sin(t)} , dt ]

[ = \ln(\sin(t)) \sin(t) - \int \cot(t) \cos(t) , dt ]

[ = \ln(\sin(t)) \sin(t) - \int \frac{\cos^2(t)}{\sin(t)} , dt ]

[ = \ln(\sin(t)) \sin(t) + \int \frac{1 - \sin^2(t)}{\sin(t)} , dt ]

[ = \ln(\sin(t)) \sin(t) + \int \frac{1}{\sin(t)} , dt - \int \sin(t) , dt ]

[ = \ln(\sin(t)) \sin(t) + \int \csc(t) , dt - (-\cos(t)) + C ]

[ = \ln(\sin(t)) \sin(t) - \ln|\csc(t) + \cot(t)| - \cos(t) + 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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