Approaching and Solving Integrals of Sine Functions: A Scientific Perspective

[BlackWyvern]

Homework Statement

What would be the best method to approach this integral? And then solve it.

$$\int\frac{dx}{(sin[x]+2)^2}$$

At first, I thought it would yield to the substitution of u = sin[x] + 2, with du = cos[x]. But this doesn't completely change the integral to one of u. Now, I don't know what else to try.

 

[Susanne217]

Homework Statement

What would be the best method to approach this integral? And then solve it.

$$\int\frac{dx}{(sin[x]+2)^2}$$

At first, I thought it would yield to the substitution of u = sin[x] + 2, with du = cos[x]. But this doesn't completely change the integral to one of u. Now, I don't know what else to try.

BlackWyvern,

Do you own one of these big Calculus books? E.g. Edwards and Penny. In the back there is an index of generalized Integrals there you will be able to find one you can use for your problem above! If not then report back to us :)

 

[BlackWyvern]

Sadly I do not. :P

I'm not too worried about an actual solution, but more how to find it. Recently I became aware of using Euler's equation to solve rational trigonometric integrals, but I have little practice. Maybe I can use this to turn the equation into something that will integrate by hand, who knows. I have seen a solution returned by Wolfram, and suffice to say... It was huge and not something that would evolve on paper. I'm pretty sure they only use heuristic algorithms, though, so I remain hopeful.

 

[Susanne217]

Sadly I do not. :P

I'm not too worried about an actual solution, but more how to find it. Recently I became aware of using Euler's equation to solve rational trigonometric integrals, but I have little practice. Maybe I can use this to turn the equation into something that will integrate by hand, who knows. I have seen a solution returned by Wolfram, and suffice to say... It was huge and not something that would evolve on paper. I'm pretty sure they only use heuristic algorithms, though, so I remain hopeful.

Look at this page here
http://en.wikipedia.org/wiki/List_of_integrals_of_rational_functions

Can you spot an integral formula you can use to solve your integral, BlackWyvern? A simular list can be found in the back of any Calculus bible...

 

[Cyosis]

Start with $u=\tan \frac{x}{2}$ and prepare yourself for a lot of work.

Edit: I noticed this is in the pre-calculus section. This type of integral (without using tables) is well above the level of pre-calc. Is this really homework or is it something you're just interested in?