How do you find the definite integral of #int (1-costheta)/(theta-sintheta)# from #[1,2]#?

Answer 1

#ln(2-sin(2))-ln(1-sin(1))~~1.9286#

Step 1. Use #u#-substitution Let #u=theta-sin(theta)# #(du)/(d theta)=(1-cos(theta))d theta#
Step 2. Use this new information to create new limits of integration #theta=1# becomes #u=1-sin(1)# #theta=2# becomes #u=2-sin(2)#
Step 3. Plug these substitutions into the integral #int_(1-sin(1))^(2-sin(2))1/u du#
Step 4. Integrate equation with #u# #int_(1-sin(1))^(2-sin(2))1/u du=[ln(u)]_(1-sin(1))^(2-sin(2))# #=ln(2-sin(2))-ln(1-sin(1))~~1.9286#
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Answer 2

To find the definite integral of ( \frac{1 - \cos(\theta)}{\theta - \sin(\theta)} ) from ( \theta = 1 ) to ( \theta = 2 ), you would typically use numerical methods, as there isn't a simple elementary antiderivative for this function. Common numerical methods include the trapezoidal rule, Simpson's rule, or numerical integration techniques implemented in software or calculators. These methods approximate the integral by dividing the interval into smaller subintervals and summing up the function values within each subinterval.

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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.

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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