How do I evaluate the integral #int(secx tanx) / (sec^2(x) - secx) dx#?

Answer 1
First of all, let's rewrite the function to integrate in a more simple way: remembering that #\sec(x)=1/\cos(x)#, and of course that #\tan(x)=\sin(x)/{\cos(x)}# we write the integrand as #\frac{\frac{1}{\cos(x)}\frac{\sin(x)}{\cos(x)}}{\frac{1}{\cos^2(x)}-\frac{1}{\cos(x)}}#, which we can simplify into #{{\sin(x)}/{\cos^2(x)}}/{{1-\cos(x)}/{\cos^2(x)}}#, and finally obtain #\frac{-\sin(x)}{\cos(x)-1}#.
To integrate this function, we'll use a couple of substitutions: first of all, by choosing #t=\cos(x)#, one has #dt=-\sin(x)\ dx#, and so #\int \frac{-\sin(x)}{\cos(x)-1} dx# becomes #\int \frac{dt}{t-1}#. By choosing #y=t-1#, one obtains #dy=dt#, and the integral becomes simply #\int \frac{dy}{y}=\log(y)+c#. Substituting back #y=t-1#, one has #\log(t-1)+c#, and again, plugging #t=\cos(x)# into the equation, one has #\log(\cos(x)-1)+c#.

By deriving, you can check that, infact, the following equation holds:

#d/dx \log(\cos(x)-1)+c = \frac{-\sin(x)}{\cos(x)-1}#

WolframAlpha for checking the derivative

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Answer 2

To evaluate the integral (\int \frac{\sec(x) \tan(x)}{\sec^2(x) - \sec(x)} , dx), you can use a substitution method. Let (u = \sec(x) - 1), then (du = \sec(x) \tan(x) , dx). Substitute (du) and (u) into the integral and simplify to solve for (u). This will allow you to integrate with respect to (u). After integrating, substitute back (u) with (\sec(x) - 1) to find the final result.

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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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