How do you find the antiderivative of #(sinx  cosx)/cosx dx#?
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where c is an arbitrary real constant.
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To find the antiderivative of ( \frac{\sin x  \cos x}{\cos x} ) with respect to ( x ):

Split the expression into two separate fractions: ( \frac{\sin x}{\cos x}  \frac{\cos x}{\cos x} ).

Rewrite ( \frac{\sin x}{\cos x} ) as ( \tan x ), since ( \tan x = \frac{\sin x}{\cos x} ).

Rewrite ( \frac{\cos x}{\cos x} ) as ( 1 ).

Now, the integral becomes ( \tan x  1 ).

Integrate ( \tan x ) and ( 1 ) separately.

The antiderivative of ( \tan x ) is ( \ln\cos x + C ), where ( C ) is the constant of integration.

The antiderivative of ( 1 ) is ( x ).

Combine the results to get the final antiderivative.
Therefore, the antiderivative of ( \frac{\sin x  \cos x}{\cos x} ) with respect to ( x ) is ( \ln\cos x + 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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