How do you find the antiderivative of #(cosx)^2/(sinx)#?
We have:
Through the Pythagorean identity:
Split the integral:
These are both wellknown integrals:
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To find the antiderivative of ( \frac{{\cos(x)^2}}{{\sin(x)}} ), we can use trigonometric identities. One approach is to rewrite the expression using the Pythagorean identity, ( \sin^2(x) + \cos^2(x) = 1 ). After rewriting, we integrate term by term using the substitution method. The steps involved are as follows:

Rewrite the expression using the Pythagorean identity: ( \frac{{\cos(x)^2}}{{\sin(x)}} = \frac{{1  \sin^2(x)}}{{\sin(x)}} ).

Expand the expression: ( \frac{{1}}{{\sin(x)}}  \frac{{\sin^2(x)}}{{\sin(x)}} ).

Rewrite each term separately: ( \csc(x)  \sin(x) ).

Integrate each term separately:
 The antiderivative of ( \csc(x) ) is ( \ln\csc(x) + \cot(x) ).
 The antiderivative of ( \sin(x) ) is ( \cos(x) ).
Therefore, the antiderivative of ( \frac{{\cos(x)^2}}{{\sin(x)}} ) is ( \ln\csc(x) + \cot(x)  \cos(x) + C ), where ( C ) is the constant of integration.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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