What is #f(x) = int cot2x dx# if #f(pi/8) = 0 #?

Answer 1

#f(x)=1/2lnsqrt2sin(2x)#

#f(x)=intcot2xdx#
#f(x)=int((cos2x)/(sin2x))dx#
now#""(d/(dx)(sin2x)=2cos2x)#

we have a log integration.

#f(x)=int((cos2x)/(sin2x))dx=1/2lnsin2x+C# #f(pi/8)=0#
#:.1/2lnsin(pi/4)+C=0#
rewrite #C=1/2lnK" "#in order to include it in the log.
#1/2lnsin(pi/4)+1/2lnk=0#
#1/2[(lnsin(pi/4)+lnk]=0#
#1/2(lnksin(pi/4))=0#
#lnx=0=>x=1#
#:. ksin(pi/4)=1#
#k(sqrt2/2)=1#
#k=2/sqrt2=sqrt2#
#f(x)=1/2lnsqrt2sin(2x)#
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Answer 2

To find ( f(x) = \int \cot^2(x) , dx ) given that ( f(\frac{\pi}{8}) = 0 ), we first need to find the antiderivative of ( \cot^2(x) ).

The antiderivative of ( \cot^2(x) ) is ( -\cot(x) ).

Using the given information, we have:

[ f(\frac{\pi}{8}) = \int_{0}^{\frac{\pi}{8}} \cot^2(x) , dx = 0 ]

Now, we can set up the integral:

[ \int_{0}^{\frac{\pi}{8}} \cot^2(x) , dx = -\cot\left(\frac{\pi}{8}\right) - (-\cot(0)) = -\cot\left(\frac{\pi}{8}\right) ]

[ -\cot\left(\frac{\pi}{8}\right) = 0 ]

Thus, ( \cot\left(\frac{\pi}{8}\right) = 0 ).

Since ( \cot\left(\frac{\pi}{8}\right) = \frac{\cos\left(\frac{\pi}{8}\right)}{\sin\left(\frac{\pi}{8}\right)} = 0 ), the numerator ( \cos\left(\frac{\pi}{8}\right) = 0 ).

Therefore, ( f(x) = \int \cot^2(x) , dx = -\cot(x) + C ), where ( C ) is the constant of integration.

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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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