Can I use only one substitution for integral?

[thereddevils]

Homework Statement

By using the substitution t = tan x, find

$$\int \frac{dx}{\cos^2 x+4\sin^2 x}$$

Homework Equations

The Attempt at a Solution

Well let tan x=t

$$\frac{dt}{dx}=\sec^2 x=\tan^2 x+1=1+t^2$$

the integral then becomes

$$\int \frac{dx}{\frac{1}{\sqrt{1+t^2}}^2+4\frac{t}{\sqrt{1+t^2}}^2}$$

which simplifies to

$$\int \frac{1}{1+4t^2}$$

Then from here i make another substitution **

let t= 1/2 tan b

dt/db = 1/2 sec^2 b

$$\int \frac{1}{1+4(\frac{1}{2} \tan b)^2} \cdot \frac{1}{2}\sec^2 b db$$

= b + constant

Back substitute

= $$\frac{1}{2}\tan^{-1} (2t)$$ + constant

= $$\frac{1}{2}\tan^{-1}(2\tan x)$$ + constant

Am i correct? Especially this part ** where i made another substitution, is that valid? Or when the question specified the substitution, i have to stick that one substitution only?

 

[CompuChip]

Looks okay to me.
If you remember that
$$\int \frac{dy}{1 + y^2} \, dy = \tan^{-1}(y)$$
then $y = 2t$ immediately gives you
$$\int \frac{dt}{1+4t^2} = \frac{1}{2} \int \frac{dy}{1 + y^2} = \frac{1}{2} \tan^{-1}(y) = \frac{1}{2} \tan^{-1}(2 \tan x)$$

(Note that if this was a definite integral, you would have to be very careful with the integration boundaries in these subsitutions)

 

[thereddevils]

Looks okay to me.
If you remember that
$$\int \frac{dy}{1 + y^2} \, dy = \tan^{-1}(y)$$
then $y = 2t$ immediately gives you
$$\int \frac{dt}{1+4t^2} = \frac{1}{2} \int \frac{dy}{1 + y^2} = \frac{1}{2} \tan^{-1}(y) = \frac{1}{2} \tan^{-1}(2 \tan x)$$

(Note that if this was a definite integral, you would have to be very careful with the integration boundaries in these subsitutions)

thanks Compuchip! So it's ok to make another substitution other than the one specified in the question? My teacher said otherwise, saying that it's not appropriate to make substitutions other than the one specified in the question. I want to get my facts right first before i get into another heated argument with her.

True, as for definite integrals, we will need to modify the limits in accordance with the substitution we make.

 

[CompuChip]

Personally, I think that if you can find a correct way to solve the question yourself (even if it is not exactly the way your teacher or anyone else has in mind) it is appropriate.

In this case, you could even argue that 2t -> t is not a full-fledged variable substitution but simply a rescaling, and the integral of 1/(1 + t²) is a standard integral which you can write down immediately whenever you encounter it (however you have proved that standard integral nicely here, which is not bad to do once in your life :) ).

 

[thereddevils]

Personally, I think that if you can find a correct way to solve the question yourself (even if it is not exactly the way your teacher or anyone else has in mind) it is appropriate.

In this case, you could even argue that 2t -> t is not a full-fledged variable substitution but simply a rescaling, and the integral of 1/(1 + t²) is a standard integral which you can write down immediately whenever you encounter it (however you have proved that standard integral nicely here, which is not bad to do once in your life :) ).

Thanks again, Compuchip.