Zeeshan 01: After a bit of research I think I can take you through this, but it'll take more than one post. If you would like that please reply. Or you could do like me and use a search engine.

Bob

]]>There are two stages to this problem. I think you have done the first but I'll say anyway.

This is a product of two functions so use the product rule:

You can combine the trig functions into a single trig function using the compound angle formulas. You want a cosine so I'll use

To make the derivative look like the right hand side here:

Hope that helps,

Bob

]]>Welcome to the forum.

Looks like you've got these sorted ; I agree with all your answers.

Bob

]]>Welcome to the forum.

I've been hoping that someone would post an answer to this and I could learn too. No such luck

So I'm having a go myself. Please note: I've never done this before, so what follows may be rubbish. Please comment …. ask for clarification … tell me why it's wrong etc. Maybe between us, we can arrive at the correct answer. And oh yes … your English is good.

So let's work with 3D coordinates with the x-y plane horizontal and z going straight up. Further, let's make the sphere have unit radius (cannot see any harm in that) and centred on the origin, O.

If P is one point of the tetrahedron, then we can specify its position using spherical coordinates theta and phi as shown here: https://en.wikipedia.org/wiki/Spherical … ate_system And let P' be the point in the x-y plane below P.

Now, what would be helpful is to have a formula for the volume of a tetrahedron in terms of theta and phi but I cannot find one. Plenty of internet pages giving the formula in vector terms such as https://math.stackexchange.com/question … ot-product

and as a determinant https://stackoverflow.com/questions/986 … n-4-points

I could also expand either into a large algebraic formula but it would take ages to enter all the LaTex so you'll have to ask nicely if you want this.

phi can take any random value from 0 to 2pi and theta any from 0 to pi.

So you can construct your function with 8 variables and there it is. Hhhmmmm.

Bob

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Sorry, but I do not understand what you are asking. Please post the whole question.

Bob

]]>Zeeshan 01 wrote:

Integration of (1/(1+sinx+cosx)) dx by letting tanx/2=z

So that's what I did.

The alternative method of post 4 does not use z at all.

You now have the presence of a function and it's derivative, so it is directly integrable as ln(function) .

Bob

]]>Try this:

https://www.mathsisfun.com/data/functio … (log (3))

Bob

]]>I used the cosine rule thus:

Bob

]]>https://www.mathsisfun.com/data/functio … )+abs(x-1)

So just show that the left and right limits are not the same.

Bob

]]>You are correct that the curve tends to zero as x tends to infinity but that doesn't tell us the range. You cannot have the value x=3; it is excluded from the domain, but you can get as close to 3 as you like and as you do the function has larger and larger values. There is no value of 'y' that cannot be attained so the range is ( -∞ , ∞ ) Note y = zero is attained at x = -2

Bob

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