- #1
dirk_mec1
- 761
- 13
Homework Statement
[tex]
\int_{0}^{\pi/4} \exp \left(-\frac{1}{\cos^2x}\right)\mbox{d}x
[/tex]
The Attempt at a Solution
Can someone give me a hint? I don't see a smart substitution nor a path via integration by parts...
yyat said:The value of the integral, whatever it is, is not going to have a simple form: The "[URL value[/URL] is not recognized by http://bootes.math.uqam.ca/cgi-bin/ipcgi/lookup.pl?Submit=GO+&number=.227651878&lookup_type=browse".
Edit: I think I overestimated the capabilities of Plouffe's inverter, so maybe the solution is simple afterall.
[tex]Dick said:Write the integral as exp(-a*sec(x)^2). Find d/da of that.
hmmm...I don't see how because:Now THAT function you can integrate dx from 0 to pi/4 (in terms of erf)..
Dick said:Not quite. The argument of the erf is t. t isn't equal to tan(x). Now keep going.
Dick said:You are getting there. There's also a 2/sqrt(pi) in the erf definition. What happened to that?
Dick said:If u=erf(sqrt(a)), what's du? Or since you've already done that u substitution for sqrt(a), if v=erf(u), what's dv?
dirk_mec1 said:Of course! The derative of erf(v) is e-v2 !
Right, I'll try again:Dick said:Dead right, but don't forget the sqrt(pi) and 2 parts of the erf definition. That's why it's actually an easy integral. But doing the v=erf(u) substitution turns the integral into v*dv, right? What's that?
dirk_mec1 said:If a = 0 then [tex]
\int_{0}^{\pi/4} \exp \left(-\frac{a}{ \cos^2x}\right)\mbox{d}x = \frac{ \pi}{4}
[/tex]
So I conclude that:
[tex] \frac{\pi}{4} = \left( \frac{- \sqrt{\pi}}{2} \right) \cdot \lim_{a \rightarrow 0} \int e^{-a} \cdot \frac{ \mbox{erf} (\sqrt{a}) }{\sqrt{a}}\ \mbox{d}a [/tex]
dirk_mec1 said:So [tex]C_0 =0[/tex]
Dick said:Sure. I did a numerical integration and I got 0.22765187804641. Does that agree with your result?
An integral is a mathematical concept that represents the area under a curve in a graph. It is used to calculate the accumulation of a quantity over a given interval.
A challenging integral is an integral that is difficult to solve using traditional methods. It may involve complex functions or require advanced techniques to evaluate.
This integral is considered challenging because it involves the exponential function and the trigonometric function cosine, which can be difficult to integrate. Additionally, the limits of integration, 0 and pi/4, make the integral more complex to solve.
Some strategies for solving this integral include using trigonometric identities, substitution, and integration by parts. It may also be helpful to break the integral into smaller, more manageable parts.
Being able to solve challenging integrals is important for advancing mathematical knowledge and solving real-world problems in fields such as physics, engineering, and economics. It also helps to develop critical thinking and problem-solving skills.