Finding equivalent forms prior to integration

[gnaritas]

I've been trying to create a program that takes a user supplied indefinite integral and integrates it using the methods I have been learning in my calculus class; substitution, integration by parts, trig substitution, partial fractions, elementary anti-derivatives, etc. I've been trying to use the MuPad computer algebra system that is part of the MatLab application. MuPad includes functions for at least integration by parts and partial fractions where I can simply input my arguments and receive the result.

I'm a new calculus student (and not particularly sharp in trigonometry or algebra either) and have some general questions about preparing to solve integration problems...

As I understand it, in an integrands initial form, a certain strategy for integration may not be available or at least readily seen, but what I would like to do is take any given integrand expression and convert it to as many alternative equivalent forms as possible to try and find integral solutions using as many different of these integration methods; substitution, IBP, trig substitution, partial fractions, as possible.

Simplify, factoring, expanding, normalize, trigonometric identities. MuPad/MatLab includes functions for performing all of these algebraic manipulations (except trig identities and maybe synthetic division.)

#1 - If I were to implement a function that would call MuPad's functions for simplify, factor, expand, synthetic division, and if I had some way of applying trig identities, would it find all of the alternative forms and allow me to determine which one allows the use as many integration strategies as possible? Are all of these functions even necessary? Is there anything wrong with using the general simplify function as opposed to functions that simplify only certain terms; radical,exponential, logarithms, sin/cos?

#2 - Do trigonometric identities act as straight substitutions of part of an expression or can they also be pathways to a new expression not directly accessible from the original expression? I was using wolframalpha.com to find alternative expressions for cos⁵x, and the only way I could find to get to their 1/16 (10 cos(x)+5 cos(3 x)+cos(5 x)) was to use both a power relations identity and a multiple-angle identity. Or is it just that the list of identities I was looking at didn't go up to cos⁵x? EDIT: Nevermind, I realize this is a stupid question.

#3 - Additionally, is there any reason for not applying the sum, difference, or constant multiple rule to an integral if it is available? What I mean to say is whether using anyone of these rules could potentially make it more difficult to integrate one of the independent integrals down the road.

#4 - Would the simplification function sometimes introduce discontinuities into my graph and prevent me from integrating?

Thanks, and I apologize if I've made any stupid misconceptions or naive assumptions about how this might work.

 

[Prove It]

I've been trying to create a program that takes a user supplied indefinite integral and integrates it using the methods I have been learning in my calculus class; substitution, integration by parts, trig substitution, partial fractions, elementary anti-derivatives, etc. I've been trying to use the MuPad computer algebra system that is part of the MatLab application. MuPad includes functions for at least integration by parts and partial fractions where I can simply input my arguments and receive the result.

I'm a new calculus student (and not particularly sharp in trigonometry or algebra either) and have some general questions about preparing to solve integration problems...

As I understand it, in an integrands initial form, a certain strategy for integration may not be available or at least readily seen, but what I would like to do is take any given integrand expression and convert it to as many alternative equivalent forms as possible to try and find integral solutions using as many different of these integration methods; substitution, IBP, trig substitution, partial fractions, as possible.

Simplify, factoring, expanding, normalize, trigonometric identities. MuPad/MatLab includes functions for performing all of these algebraic manipulations (except trig identities and maybe synthetic division.)

#1 - If I were to implement a function that would call MuPad's functions for simplify, factor, expand, synthetic division, and if I had some way of applying trig identities, would it find all of the alternative forms and allow me to determine which one allows the use as many integration strategies as possible? Are all of these functions even necessary? Is there anything wrong with using the general simplify function as opposed to functions that simplify only certain terms; radical,exponential, logarithms, sin/cos?

#2 - Do trigonometric identities act as straight substitutions of part of an expression or can they also be pathways to a new expression not directly accessible from the original expression? I was using wolframalpha.com to find alternative expressions for cos⁵x, and the only way I could find to get to their 1/16 (10 cos(x)+5 cos(3 x)+cos(5 x)) was to use both a power relations identity and a multiple-angle identity. Or is it just that the list of identities I was looking at didn't go up to cos⁵x? EDIT: Nevermind, I realize this is a stupid question.

#3 - Additionally, is there any reason for not applying the sum, difference, or constant multiple rule to an integral if it is available? What I mean to say is whether using anyone of these rules could potentially make it more difficult to integrate one of the independent integrals down the road.

#4 - Would the simplification function sometimes introduce discontinuities into my graph and prevent me from integrating?

Thanks, and I apologize if I've made any stupid misconceptions or naive assumptions about how this might work.

$$\displaystyle \int{\cos^5{x}\,dx} = \int{\cos^4{x}\cos{x}\,dx}$$

$$\displaystyle = \int{(\cos^2{x})^2\cos{x}\,dx}$$

$$\displaystyle = \int{(1 - \sin^2{x})^2\cos{x}\,dx}$$.

Now make the substitution $$\displaystyle u = \sin{x}$$ so that $$\displaystyle \frac{du}{dx} = \cos{x}$$ and the integral becomes

$$\displaystyle \int{(1 - u^2)^2\,\frac{du}{dx}\,dx} = \int{(1 - u^2)^2\,du}$$

$$\displaystyle = \int{1 - 2u^2 + u^4\,du}$$

$$\displaystyle = u - \frac{2u^3}{3} + \frac{u^5}{5} + C$$

$$\displaystyle = \sin{x} - \frac{2\sin^3{x}}{3} + \frac{\sin^5{x}}{5} + C$$.P.S. There is something wrong with the LaTeX on this site - it appears that once something is compiled it can't be deleted or edited.

 

[Mark44]

P.S. There is something wrong with the LaTeX on this site - it appears that once something is compiled it can't be deleted or edited.

This is a well-known problem. The LaTeX script winds up in the browser's local cache. When you revise the LaTeX script, what gets rendered is what was already in the cache. The workaround is to click the browser's refresh button.

 

[gnaritas]

$$\displaystyle \int{\cos^5{x}\,dx} = \int{\cos^4{x}\cos{x}\,dx}$$

$$\displaystyle = \int{(\cos^2{x})^2\cos{x}\,dx}$$

$$\displaystyle = \int{(1 - \sin^2{x})^2\cos{x}\,dx}$$.

Now make the substitution $$\displaystyle u = \sin{x}$$ so that $$\displaystyle \frac{du}{dx} = \cos{x}$$ and the integral becomes

$$\displaystyle \int{(1 - u^2)^2\,\frac{du}{dx}\,dx} = \int{(1 - u^2)^2\,du}$$

$$\displaystyle = \int{1 - 2u^2 + u^4\,du}$$

$$\displaystyle = u - \frac{2u^3}{3} + \frac{u^5}{5} + C$$

$$\displaystyle = \sin{x} - \frac{2\sin^3{x}}{3} + \frac{\sin^5{x}}{5} + C$$.


P.S. There is something wrong with the LaTeX on this site - it appears that once something is compiled it can't be deleted or edited.

My question about cos⁵x was about trig identities, not about how it is integrated, but I realize now it was kind of a stupid question, and they just used a power reduction identity for cos⁵x.