# Integral problems

mansi
i think i don't have adequate theory to solve these problems...

1.given that f(x) =cos(x) sin^k(x) / (1+x). calculate integral of this function wrt x between limits 0 and pi/2 . then find the it's limit as k tends to infinity...

2.let f(x)={e^(-ax)-e^(-bx)}/x, 0< a< b .let I be the integral of f wrt x between 0 and infinity. express I as a double integral...

kleinwolf
1) Are you meaning : $$\sin^k(x)\textrm{ or }\sin(x)^k$$ ?

mansi
the former...

Homework Helper
Do you have to compute these 2

$$F(k)=:\int_{0}^{\frac{\pi}{2}}\left[\cos x\left(\frac{\sin^{k}x}{1+x}\right)\right] dx$$

$$\lim_{k\rightarrow +\infty} F(k)$$

...?

Daniel.

Homework Helper

$$F(1)=\frac {1}{2}\mbox{Si}\left( 2+\pi \right) \cos 2-\frac {1}{2}\mbox{Ci}\left( 2+\pi \right) \sin 2-\frac {1}{2}\mbox{Si}\left( 2\right) \cos 2+ \frac {1}{2}\mbox{Ci}\left( 2\right) \sin 2$$

$$F(2)= -\frac{1}{4}\mbox{Si}\left( \frac {3}{2}\pi +3\right) \sin 3-\frac{1}{4}\mbox{Ci}\left( \frac {3}{2}\pi +3\right) \cos 3+\frac{1}{4}\mbox{Si}\left( 1+\frac{1}{2}\pi \right) \sin 1$$
$$+\frac{1}{4}\mbox{Ci}\left( 1+\frac{1}{2}\pi \right) \cos 1+\frac{1}{4}\mbox{Si}\left( 3\right) \sin 3+\frac{1}{4}\mbox{Ci}\left( 3\right) \cos 3-\frac{1}{4}\mbox{Si}\left( 1\right) \sin 1-\frac{1}{4}\mbox{Ci}\left( 1\right) \cos 1$$

Daniel.

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Homework Helper
Here you can compute other values of F.I computed $F(78)$.

Daniel.

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kleinwolf
Well the first is really unsolvable I think...even the latter is hard...maybe a start like this :

$$\frac{sin(x)^kcos(x)}{1+x}=\left(\frac{e^{ix}-e^{-ix}}{2i}\right)^k\frac{e^{ix}+e^{-ix}}{2}\frac{1}{1+x}$$

$$= \frac{1}{2^{k+1}i^k}\sum_{n=0}^k\left(\begin{array}{c} k\\n\end{array}\right)(-1)^{n-k}\frac{e^{i(2n-k+1)x}}{1+x}+\frac{e^{i(2n-k-1)x}}{1+x}$$

Remain to calculate things of the form $$\int_0^{\frac{\pi}{2}}\frac{e^{imx}}{1+x}=\int_0^{\frac{\pi}{2}}e^{imx}\sum_{p=0}^\infty(-1)^p x^p$$

...well no sorry, i get lost and wrong, maybe some contour integral with residues in the complex plane...or a recursion relation over k

Homework Helper
I linked you to a website which uses WebMathematica.I used my own antique Maple to come up with this (see attached file).

I can only assume that the limit is 0...

Daniel.

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Homework Helper
1. Do you need to find the integral exactly? A bound will suffice if it's just the limit you're interested in, 1/(1+x) is bounded by 1 on [0..pi/2], cos and sin are non-negative there.

Homework Helper
I've messed about in mathematica and got that:

$$\int_0^{\frac{\pi}{2}} \frac{\cos x \sin^{1000000000} x}{1 + x} dx = 0 \quad \text{(to 100 dp)}$$

Also if you look at the graph of the function as you increase k for either the whole function or just sink (x) it starts to look like the area does tend towards 0.

Perhaps this is about looking at the properties of the function, it would seem that:

$$\lim_{k \rightarrow \infty} \sin^k x = \left\{ \begin{array}{rcl} 1 & \mbox{for} & x = \pi \left(2n + \frac{1}{2} \right) \quad n \in \mathbb{Z}\\ \text{Undefined} & \mbox{for} & x = \pi \left(2n - \frac{1}{2} \right) \quad n \in \mathbb{Z} \\ 0 & \mbox{for} & \text{elsewhere} \end{array}\right.$$

Now using that to look at the original function it's fairly easy to derive that:

$$\lim_{k \rightarrow \infty} \frac{\cos x \sin^{k} x}{1 + x} = 0 \quad \forall x \in \left[0,\frac{\pi}{2}\right]$$

Suddenly looks a lot easier to integrate to me

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mansi
thanks for the help...
never mind the second one...i figured it out.

kleinwolf
Do you know why we always write sin(x)^k=sin^k(x) ?

This seems to me a notation abuse. Let's take k positive integer, then

sin(x)^k=sin(x)*sin(x)*...sin(x) k times

sin^k(x)=sin(sin(...sin(x)))...)) k times iterated

Or you have another notation for iteration n-times ?

mansi
sin^k(x)=(sinx)^k =sinx *sinx*...k times
and to express the second one let f(x)=sinx
then f^k(x)= (sin(sin(...sin(x)))...))

Homework Helper
Actually it's not to confuse $\sin (x^k)$ with $(\sin x)^k$ as most people don't bother writing out the brackets.

But yeah notations don't seem to be entirely generalized.

whozum
klein, I've never seen the sin function self-iterated like you demonstrated,

$$sin^k(x) = sin(x)^k$$

The one on the RHS isn't used because of this one,
and they arent equalto $$sin(x^k)$$

Homework Helper
Actually the argument of $\sin x$ is never paranthesized,unless it gets really complicated...There's no point in writing $\sin\left(x\right)$,but it is for $\sin\left(x+y\right)$.

Daniel.

Homework Helper
dextercioby said:
Actually the argument of $\sin x$ is never paranthesized,unless it gets really complicated...There's no point in writing $\sin\left(x\right)$,but it is for $\sin\left(x+y\right)$.

Daniel.
But that's my point, can you see the difference between:

$$\sin {x^k}$$

And:

$${\sin x}^k$$

Unless you look at the code?

Homework Helper
I didn't look.Did u use "\displaystyle" ...?

Daniel.

EDIT:No,you didn't. :tongue2: I'm not trying to impose my own beliefs,just to stress out that writing

$$\sin x^{k}\neq \left(\sin x\right)^{k}\equiv \sin^{k} x$$

has a motivation.If someone else sees it and agrees on it,then I'm happy...

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Moo Of Doom
But it does bother me that $$\sin^2{x}=(\sin{x})^2$$ but $$\sin^{-1}{x}\neq \frac{1}{\sin{x}}$$! Why use the same notation for these two concepts? Why does the k in sin^k(x) represent a functional power if k=-1, but not for any other k? Bah!

whozum
Wonder what $sin^{-2}x$ would translate as?

Homework Helper
whozum said:
Wonder what $sin^{-2}x$ would translate as?
$$sin^{-2}x = \frac{1}{(\sin x)^2}$$

The notation may be odd but it's fairly common and well known.

Moo Of Doom
Hmm... on that note, what do you think $$\sin^0{x}$$ is? 1 or x? I vote for 1.

whozum
Well technically only $sin^{-1}x$ is the arcsin function, so how about

$$sin^{(-1)(0)}x$$

$$arcsin(x)^0 \ or \ 1^{-1}$$

Hmm... on that note, what do you think $$\sin^0{x}$$ is? 1 or x? I vote for 1.
You'd be right, apart from when $x = k\pi \quad k \in \mathbb{Z}$. Look it's really simple in general:
$$\sin^a x = \left\{ \begin{array}{rcl} \arcsin x & \mbox{for} & x = -1\\ (\sin x)^a & \mbox{for} & \text{elsewhere} \\ \end{array}\right.$$