Question

Note: This is NOT a question. This is a tutorial.

How to integrate using the Wallis' Formula? See comment below to see how!

Answer 1

How to integrate using the Wallis' Formula? See comment below to see how!

The Wallis' Formula can ONLY be done if the integral is in the form:

\(\LARGE \int_{0}^{\frac{\pi}{2}} (\sin^m u \cos^n u) du\)

Where:
m, n \(\ge\) 0

u = 0; \(\large u_{2} = \frac{\pi}{2}\)

Note that the limits should be from 0 to \(\frac{\pi}{2}\) and the integrand should contain sine and cosine ONLY.


When it is already in this form, then you can integrate it by using this formula:

\(\Large \frac{[(m-1)(m-3)(m-5)..... \stackrel{2}{1}][(n-1)(n-3)(n-5).... \stackrel{2}{1}]}{[(m+n)(m+n - 2)(m+n-4)..... \stackrel{2}{1}]}\)

looks complicated? it's really just simple! What do these mean? It's like getting the factorial of something. but rather than constantly subtracting 1, you subtract by the odd numbers 1, 3, 5, etc. until you reach 2 or 1. Do that with m and n then multiply. Same thing with the denominator. Add the exponents of your sine and cosine, then keep on subtracting by even numbers.

Example:

\(\LARGE \int_{0}^{\frac{\pi}{2}} (\sin^15 x \cos^14 x) dx\)

Here, our m is 15, and our n is 14. so we do the Wallis Formula

(m-1) = 14
(m-3) = 12
and so on...

so we have...

\(\Large \frac{[(14)(12)(10)(8)6)(4)(2)][(n-1)(n-3)(n-5).... \stackrel{2}{1}]}{[(m+n)(m+n - 2)(m+n-4)..... \stackrel{2}{1}]}\)


We reached 2 so we stop there..we do n now.

n-1 is 13
n-2 is 11
and so on...

so we have...

\(\Large \frac{[(14)(12)(10)(8)6)(4)(2)][(13)(11)(9)(7)(5)(3)(1)}{[(m+n)(m+n - 2)(m+n-4)..... \stackrel{2}{1}]}\)

we reached 1 so we stop there..now we do the denominator

m+ n = 29 so...

\(\Large \frac{[(14)(12)(10)(8)6)(4)(2)][(13)(11)(9)(7)(5)(3)(1)}{[(29)(27)(25)(23)(21)(19)(17)(15)(13)(11)(9)(7)(5)(3)(1)]}\)

as you can see there is something we can cancel...


\(\Large \frac{[(14)(12)(10)(8)6)(4)(2)][\cancel{(13)(11)(9)(7)(5)(3)(1)}]}{[(29)(27)(25)(23)(21)(19)(17)(15)\cancel{(13)(11)(9)(7)(5)(3)(1)}]}\)

so now we only have

\(\Large \frac{[(14)(12)(10)(8)6)(4)(2)]}{[(29)(27)(25)(23)(21)(19)(17)(15)]}\)

simplifying that we have

\(\Large \frac{645120}{4.58 \times 10^{10}}\)

Answer 2

@Mimi_x3 something new and interesting >:))

Answer 3

how long did it take you to write all of this lol?

Answer 4

idk 30 minutes?

Answer 5

What exactly is the purpose of this?

Answer 6

do you mean the method? or the tutorial?

Answer 7

to help us lol

Answer 8

uhmm wait there's a latex malfunction in the given i just noticed...the problem is

\(\LARGE \int_{0}^{\frac{\pi}{2}} (\sin^{15} x \cos^{14} x)dx\)

Answer 9

the latex looks cool <3 haha

Answer 10

Uhm, I wanted to learn what the Wallis' formula was :/

Answer 11

Awesome tutorial ,@lgbasallote ..Ty :)

Answer 12

nice
can u differentiate root x?

Answer 13

thanks @eyad ^_^ glad to help :D

Answer 14

@Ruchi yes i can :D

Answer 15

i hav posted it.