integral of sin x using the definition

Hi,

I am trying to calculate:

using the definition only, that is, that , where . If we take take we obtain:

.

I am stuck at this point. Is there any kind of identity, perhaps trigonometric or power series related, to help me evaluate this limit by hand only?

Thank in advanced or any help,

James

Re: integral of sin x using the definition

Quote:

Originally Posted by

**james121515** Hi,

I am trying to calculate:

using the definition only, that is, that

, where

. If we take take

we obtain:

.

I am stuck at this point. Is there any kind of identity, perhaps trigonometric or power series related, to help me evaluate this limit by hand only?

Thank in advanced or any help,

James

Surely you should know that by definition...

Re: integral of sin x using the definition

Quote:

Originally Posted by

**james121515** Hi,

I am trying to calculate:

using the definition only, that is, that

, where

. If we take take

we obtain:

.

I am stuck at this point. Is there any kind of identity, perhaps trigonometric or power series related, to help me evaluate this limit by hand only?

Thank in advanced or any help,

James

Hint:

Use Euler's formula - Wikipedia, the free encyclopedia.

Edit:

An alternative:

Now multiply both sides by .

And third alternative:

Proving by induction:

Re: integral of sin x using the definition

Quote:

Originally Posted by

**Prove It** Surely you should know that

by definition...

I'm not sure it's a definition. I suspect the OP knows the rule and wants to derive the rule from first principles.

Re: integral of sin x using the definition

Re: integral of sin x using the definition

Following previous outlines and denoting

If , we have:

Identifying imaginary parts

So,

Then,

Re: integral of sin x using the definition

Quote:

Originally Posted by

**chisigma** Using the 'identity'...

(2)

Thanks for your reply. I had no idea the solution to this involves complex analysis. This first 'identity' you mention here, is this derived from Euler's formula or is it more involved?

Re: integral of sin x using the definition

Quote:

Originally Posted by

**james121515** Thanks for your reply. I had no idea the solution to this involves complex analysis. This first 'identity' you mention here, is this derived from Euler's formula

or is it more involved?

Yes. And the sum of a geometric series is being used.