# How do you evaluate the integral #int sinthetaln(costheta)#?

If we substitute:

we have that:

We can calculate this integral by parts:

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To evaluate the integral ∫ sin(θ) ln(cos(θ)) dθ, we can use integration by parts.

Let u = ln(cos(θ)) and dv = sin(θ) dθ. Then, du = -tan(θ) dθ and v = -cos(θ).

Applying the integration by parts formula: ∫ u dv = uv - ∫ v du,

we get: ∫ sin(θ) ln(cos(θ)) dθ = -ln(cos(θ)) * cos(θ) - ∫ -cos(θ) * (-tan(θ)) dθ.

Simplify and integrate the remaining integral: = -cos(θ) ln(cos(θ)) + ∫ cos(θ) tan(θ) dθ.

Now, we have another integral to evaluate, which can be done by using substitution method or recognizing it as a known integral.

After evaluating the integral, you will obtain the final result.

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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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