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### Integral of $\sqrt{1+\tan^2x}$ [duplicate]

Possible Duplicate: Ways to evaluate $\int \sec \theta d \theta$ I'm having a bit of a problem with an integral. The original problem was the length of a curve given parametrically. I've managed ...
1k views

### Proof of formula for the antiderivative of $\sec x$ [duplicate]

How do I prove that indefinite integral of $\sec x$ is equal to $\ln(\sec x + \tan x) + C$? I tried to substitute $t = \cos x$ but that didn't help. I have no idea how to integrate it any other way, ...
242 views

### Evaluating $\int\sec x \,\mathrm dx$ [duplicate]

$$\int\sec x \,\mathrm dx = \ln\left|\sec{x} + \tan{x}\right|+ C = \ln{\left|\tan\left(\frac{x}{2} + \frac{\pi}{4}\right)\right|} + C$$ My question is how? How are these derived?
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### Why is $\int \sec x\,dx$ equal to $\ln | \sec x + \tan x |$? [duplicate]

Everyone knows that $\int \sec x\,dx = \ln | \sec x + \tan x |$? But how to reach it through a conscious deduction, through a clear and objective way?
157 views

### Find integral $\int\frac{1}{\cos x}dx$ [duplicate]

Need help with this integral $$\int\frac{1}{\cos x}dx$$ I know that the answer is $$\ln|\operatorname{tg} x+\sec x|$$ I tried transforming 1 into $\cos^2x + \sin^2x$ but it led to nothing. Need to ...
60 views

### Arriving to the integral of $\sec(x)$ [duplicate]

So I was looking at how to obtain the following integral $$\int \sec(x)\,\mathrm dx$$ I saw that, to solve it, they divided and multiplied by $\tan(x)sec(x)/tan(x)sec(x)$. I understand that this is ...
66 views

### Another method to integrate $\int\sec x dx?$ [duplicate]

$$\int\sec x dx?$$ This is the method I used: $$\int\sec x\times\frac{\sec x +\tan x}{\sec x +\tan x}dx=\int\frac{\sec^2x+\sec x\tan x}{\sec x +\tan x}=\ln(\sec x +\tan x)+c$$ I would like to see ...
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### List of interesting integrals for early calculus students

I am teaching Calc 1 right now and I want to give my students more interesting examples of integrals. By interesting, I mean ones that are challenging, not as straightforward (though not extremely ...
5k views

### Graphs for which a calculus student can reasonably compute the arclength

Given a differentiable real-valued function $f$, the arclength of its graph on $[a,b]$ is given by $$\int_a^b\sqrt{1+\left(f'(x)\right)^2}\,\mathrm{d}x$$ For many choices of $f$ this can be a ...