Linked Questions

3
votes
3answers
23k views

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 ...
4
votes
4answers
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, ...
2
votes
3answers
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?
1
vote
2answers
134 views

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?
0
votes
3answers
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 ...
0
votes
3answers
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 ...
0
votes
5answers
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 ...
73
votes
16answers
30k views

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 ...
60
votes
8answers
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 ...
17
votes
5answers
3k views

Integration of secant

$$\begin{align} \int \sec x \, dx &= \int \cos x \left( \frac{1}{\cos^2x} \right) \, dx \\ &= \int \cos x \left( \frac{1}{1-\sin^2x} \right) \, dx \\ & = \int\cos x\cdot\frac{1}{1-\...
14
votes
7answers
2k views

Solve trigonometric equation: $1 = m \; \text{cos}(\alpha) + \text{sin}(\alpha)$

Dealing with a physics Problem I get the following equation to solve for $\alpha$ $1 = m \; \text{cos}(\alpha) + \text{sin}(\alpha)$ Putting this in Mathematica gives the result: $a==2 \text{ArcTan}...
20
votes
2answers
21k views

How to integrate $ \int \frac{x}{\sqrt{x^4+10x^2-96x-71}}dx$?

I read about $ \int \dfrac{x}{\sqrt{x^4+10x^2-96x-71}}dx$ on the Wikipedia Risch algorithm page. They gave an answer but I don't understand how how they got it.
8
votes
4answers
3k views

Integrate ${\sec 4x}$

How do I go about doing this? I try doing it by parts, but it seems to work out wrong: $\eqalign{ & \int {\sec 4xdx} \cr & u = \sec 4x \cr & {{du} \over {dx}} = 4\sec 4x\tan 4x ...
3
votes
2answers
6k views

Integral related to Arcsinh(x)

It is not difficult to verify that $$ \frac{\mathrm d}{\mathrm dx} \left[ \log\Big(x+\sqrt{x^2+1}\Big) \right] = \frac{1}{\sqrt{1+x^2}} $$ for $x\geq 0$, say. How would one calculate the indefinite ...
5
votes
4answers
236 views

Is there a more elegant way of computing $\int \frac{1}{\sin(x)}dx$ and $\int \frac{1}{\cos(x)}dx$?

Both integrals can be solved by substitution, and while I am comfortable with that, in both cases I find the method unbearably ugly, mostly because there are hundreds of overtly feasible substitutions ...

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