Integration by u substitution for inverse trig formulas

[LearninDaMath]

Homework Statement

You know the U substitution proofs for inverse trig functions that go like this:

\int\frac{1}{a^{2}+x^{2}}dx

\int\frac{a\frac{1}{a}}{a(1+\frac{x^2}{a^2})}dx

let u = x/a

du= dx/a

...

\frac{1}{a}tan^{-1}(x/a)+cI have searched google and can't find any of these proofs for the other trigonometric functions like sin and tan. Where can I find step by steps of these particular integration generalizations?

Do these U substitution methods for inverse trig functions have some sort of specific name?

 

[HallsofIvy]

I'm surprised that you weren't able to find them. try
http://www.millersville.edu/~bikenaga/calculus/invtrig/invtrig.html and
http://math.usask.ca/maclean/124/Inverse_Trig_Functions.pdf

 

[LearninDaMath]

Hi Hallsofivy, really appreciate it. However, of the links you kindly provided, there is only one instance of a formula of which I'm describing. One for arcsin. But there seems to be no resource on the web that displays all or even 2 or 3 of these particular formulas. I've searched every combination of "inverse," "trigonometric," "integration," "subsititution," "formula(s)." I don't know how else to describe these specific formulas. Do you have any other keyword suggestions or links that provide on this integration topic?

 

[SammyS]

Homework Statement

You know the U substitution proofs for inverse trig functions that go like this:

\int\frac{1}{a^{2}+x^{2}}dx

\displaystyle \int\frac{\not{a}\frac{1}{a}}{a(1+\frac{x^2}{a^2})}dx

I have crossed out an extra letter a , which you had in the numerator.

let u = x/a

du= dx/a
...

\frac{1}{a}tan^{-1}(x/a)+c

I have searched google and can't find any of these proofs for the other trigonometric functions like sin and tan. Where can I find step by steps of these particular integration generalizations?

Do these U substitution methods for inverse trig functions have some sort of specific name?

I don't see how what you have is much of a proof of anything.

You can get this from taking the derivative of tan^-1(x/a).

\displaystyle \frac{d}{dx}\tan^{-1}\left(\frac{x}{a}\right)=\frac{1}{a}\frac{1}{(x/a)^2+1}​

To do something similar with arcsin & arccos, go ahead and take the derivatives:

\displaystyle \frac{d}{dx}\sin^{-1}\left(\frac{x}{a}\right)

\displaystyle \frac{d}{dx}\cos^{-1}\left(\frac{x}{a}\right)

 

[LearninDaMath]

I have crossed out an extra letter a , which you had in the numerator.

I don't see how what you have is much of a proof of anything.

You can get this from taking the derivative of tan^-1(x/a).

\displaystyle \frac{d}{dx}\tan^{-1}\left(\frac{x}{a}\right)=\frac{1}{a}\frac{1}{(x/a)^2+1}​

To do something similar with arcsin & arccos, go ahead and take the derivatives:

\displaystyle \frac{d}{dx}\sin^{-1}\left(\frac{x}{a}\right)

\displaystyle \frac{d}{dx}\cos^{-1}\left(\frac{x}{a}\right)

I didn't mean proof, sorry about that. I guess I just meant formula. Thanks, I'll give it a try.

 

[LearninDaMath]

sin^{-1}\frac{x}{a}=y

\frac{d}{dx}sin^{-1}=\frac{d}{dx}\frac{x}{a}

cos(y)y'= \frac{1}{a}

y'=\frac{1}{acos(y)}

y'=\frac{1}{acos(sin^{-1}(\frac{x}{a})^{2})}

Trig Identity: sin^{2}y+cos^{2}y=1

cosy=\sqrt{1-(\frac{x}{a})^2}

y'=\frac{1}{acos(y)}=\frac{1}{a\sqrt{1-(\frac{x}{a})^{2}}}So when I do the derivative of arcsine, I get \frac{d}{dx}sin^{-1}\frac{x}{a}=\frac{1}{a\sqrt{1-(\frac{x}{a})^{2}}}

However, the formula states:

\int\frac{1}{\sqrt{a^2-x^2}}dx=sin^{-1}\frac{x}{a}+C

I seem to have gotten an "a" outside of the radical and a "1" inside the radical, while the formula shows nothing outside the radical and an "a^{2} inside the radical. Where might I have gone off track to arrive at a final answer with misarranged variables?

EDIT:

I think I see it, would it just be a matter of distributing the "a" into the radical?

 

[SammyS]

...

So when I do the derivative of arcsine, I get \frac{d}{dx}sin^{-1}\frac{x}{a}=\frac{1}{a\sqrt{1-(\frac{x}{a})^{2}}}

However, the formula states:

\int\frac{1}{\sqrt{a^2-x^2}}dx=sin^{-1}\frac{x}{a}+C

I seem to have gotten an "a" outside of the radical and a "1" inside the radical, while the formula shows nothing outside the radical and an "a^{2} inside the radical. Where might I have gone off track to arrive at a final answer with misarranged variables?

EDIT:

I think I see it, would it just be a matter of distributing the "a" into the radical?

Yes, they are equivalent.

 

[LearninDaMath]

Thanks SammyS,

I have crossed out an extra letter a , which you had in the numerator.

I don't see how what you have is much of a proof of anything.

You can get this from taking the derivative of tan^-1(x/a).

\displaystyle \frac{d}{dx}\tan^{-1}\left(\frac{x}{a}\right)=\frac{1}{a}\frac{1}{(x/a)^2+1}​

To do something similar with arcsin & arccos, go ahead and take the derivatives:

\displaystyle \frac{d}{dx}\sin^{-1}\left(\frac{x}{a}\right)

\displaystyle \frac{d}{dx}\cos^{-1}\left(\frac{x}{a}\right)

Also,

You show that the derivative of arctanx is:

\displaystyle \frac{d}{dx}\tan^{-1}\left(\frac{x}{a}\right)=\frac{1}{a}\frac{1}{(x/a)^2+1}​

But if I am interested in deriving the function \frac{1}{a}tan^{-1}(\frac{x}{a}) for this particular formula, does the extra fraction \frac{1}{a} that the function to be derived includes account for the final result of \frac{1}{a}\frac{1}{a}\frac{1}{(\frac{x}{a})^{2}+1} = \frac{1}{x^{2}+a^{2}}?

 

[SammyS]

Thanks SammyS,
Also,

You show that the derivative of arctanx is:

\displaystyle \frac{d}{dx}\tan^{-1}\left(\frac{x}{a}\right)=\frac{1}{a}\frac{1}{(x/a)^2+1}​

But if I am interested in deriving the function \frac{1}{a}tan^{-1}(\frac{x}{a}) for this particular formula, does the extra fraction \frac{1}{a} that the function to be derived includes account for the final result of \frac{1}{a}\frac{1}{a}\frac{1}{(\frac{x}{a})^{2}+1} = \frac{1}{x^{2}+a^{2}}?

The algebra says yes.