# What is #f(x) = int sin^2x-cotx dx# if #f((5pi)/4) = 0 #?

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To solve the integral ( f(x) = \int \sin^2(x) - \cot(x) , dx ) given that ( f\left(\frac{5\pi}{4}\right) = 0 ), we need to evaluate the integral and then find the constant of integration using the given condition.

Integrating ( \sin^2(x) - \cot(x) ) with respect to ( x ) yields: [ \int \sin^2(x) - \cot(x) , dx = \frac{x}{2} + \frac{\cos(2x)}{4} - \ln|\sin(x)| + C ]

Now, using the condition ( f\left(\frac{5\pi}{4}\right) = 0 ), we substitute ( x = \frac{5\pi}{4} ) into the expression for ( f(x) ) and solve for the constant of integration ( C ):

[ 0 = \frac{\frac{5\pi}{4}}{2} + \frac{\cos\left(\frac{5\pi}{2}\right)}{4} - \ln|\sin\left(\frac{5\pi}{4}\right)| + C ]

[ 0 = \frac{5\pi}{8} + \frac{0}{4} - \ln\left|\frac{\sqrt{2}}{2}\right| + C ]

[ 0 = \frac{5\pi}{8} - \frac{\ln(\sqrt{2}/2)}{4} + C ]

[ C = \frac{\ln(\sqrt{2}/2)}{4} - \frac{5\pi}{8} ]

Thus, the function ( f(x) ) is: [ f(x) = \frac{x}{2} + \frac{\cos(2x)}{4} - \ln|\sin(x)| + \frac{\ln(\sqrt{2}/2)}{4} - \frac{5\pi}{8} ]

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To find ( f(x) = \int \sin^2(x) - \cot(x) , dx ) when ( f\left(\frac{5\pi}{4}\right) = 0 ), we first integrate the function:

[ \int \sin^2(x) - \cot(x) , dx ]

[ = \int \sin^2(x) - \frac{\cos(x)}{\sin(x)} , dx ]

[ = \int \sin^2(x) - \frac{1}{\tan(x)} , dx ]

[ = \int \sin^2(x) - \frac{\sin(x)}{\cos(x)} , dx ]

[ = \int \sin^2(x) - \tan(x) , dx ]

Now, integrate term by term.

[ = \left( -\frac{\cos(2x)}{4} + \ln|\sin(x)| \right) + C ]

Given that ( f\left(\frac{5\pi}{4}\right) = 0 ), we substitute ( x = \frac{5\pi}{4} ) into the integrated function and set it equal to 0:

[ 0 = -\frac{\cos\left(\frac{5\pi}{2}\right)}{4} + \ln|\sin\left(\frac{5\pi}{4}\right)| + C ]

[ 0 = -\frac{0}{4} + \ln\left|\frac{\sqrt{2}}{2}\right| + C ]

[ 0 = 0 + \ln\left(\frac{\sqrt{2}}{2}\right) + C ]

[ C = -\ln\left(\frac{\sqrt{2}}{2}\right) ]

So, the function ( f(x) = \int \sin^2(x) - \cot(x) , dx ) with the condition ( f\left(\frac{5\pi}{4}\right) = 0 ) is:

[ f(x) = -\frac{\cos(2x)}{4} + \ln|\sin(x)| - \ln\left(\frac{\sqrt{2}}{2}\right) ]

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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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