How do you find the antiderivative of #cos(x)/(1-cos(x))#?

Answer 1

#-x-2cot(x/2)+C#

#I=intcos(x)/(1-cos(x))dx#

Rewriting the integral in a simpler form:

#I=int((cos(x)-1)+1)/(1-cos(x))dx#

#I=int(-(1-cos(x)))/(1-cos(x))dx+intdx/(1-cos(x))#

#I=-intdx+intdx/(1-cos(x))#

#I=-x+intdx/(1-cos(x))#

For the remaining integral, we'll use the tangent half-angle substitution which uses #t=tan(x/2)#. The function #cos(x)# can be expressed in terms of #tan(x/2)# as follows:

That is, #cos(x)=(1-tan^2(x/2))/(1+tan^2(x/2))=(1-tan^2(x/2))/sec^2(x/2)#.

Also note that the substitution #t=tan(x/2)# implies #dt=1/2sec^2(x/2)dx#.

Applying this to the integral gives:

#I=-x+intdx/((1-(1-tan^2(x/2))/sec^2(x/2)))#

#I=-x+int(sec^2(x/2)dx)/(sec^2(x/2)-(1-tan^2(x/2))#

Rewriting #sec^2(x/2)# as #tan^2(x/2)+1#:

#I=-x+int(sec^2(x/2)dx)/(2tan^2(x/2))#

#I=-x+2int(1/2sec^2(x/2)dx)/(2tan^2(x/2))#

Substituting:

#I=-x+2intdt/t^2#

#I=-x-2/t+C#

#I=-x-2/tan(x/2)+C#

#I=-x-2cot(x/2)+C#

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Answer 2

#-(x+cot(x/2) + C#

#int cos x/(1-cos x) dx=int (1-2sin^2(x/2))/(2sin^2(x/2)) dx#
#=int1/2 csc^2(x/2) dx-int dx#
#=intd(-cot(x/2) -x#
#=-(x+cot(x/2)) + C#.
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Answer 3

To find the antiderivative of cos(x)/(1-cos(x)), you can use a trigonometric substitution. Let u = 1 - cos(x). Then, du = sin(x) dx. Rewrite the integral in terms of u: ∫ du/u. This integral evaluates to ln|u| + C, where C is the constant of integration. Finally, substitute back u = 1 - cos(x) to get the final result: ln|1 - cos(x)| + C.

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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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