Math Substitution for Solving Equations | Attached Picture Included

[COCoNuT]

the problem is shown in the picture i attached.

wouldnt i use u = 2sec(\theta) as a sub?

 

[HallsofIvy]

Well, not "u= 2 sec θ" because there was no "u" in the integral!

But setting 2x= secθ will work very nicely (Notice where the 2 is!)

sin²θ+ cos²θ= 1 so, dividing both sides by cos²θ, tan²θ+ 1= sec²θ and then
sec²θ- 1= tan²θ.

That's the whole point of the substitution. If 2x= sec θ then 4x² =
sec² θ so 4x²- 1= tan²θ and you can drop the square root: \sqrt{4x^2-1}= \sqrt{tan^2\theta}= tan \theta.

(You might have been thinking of examples involving "x²- 4" where the substitution x= 2sec θ would give 4sec²- 4= 4(tan² θ)

 

[COCoNuT]

ok so the problem should look like \int \frac{x^3}{tan \theta} right now.

i don't know what to do next. i can't use u-du. can i use parts on this?

if you set 2x= sec \theta

then x= \frac{sec \theta}{2}
then would i have to take the derv. of that?
dx= 1/2*ln(sec(\theta)+tan(\theta))


then we would rewrite the problem as...(plugged in x=\frac{sec \theta}{2} for x)

\int\frac{(\frac{sec \theta}{2})^3}{\sqrt{4(\frac{sec \theta}{2})^2-1}} * 1/2*ln(sec(\theta)+tan(\theta))

hmm my way seems longer and harder...


can you help me with your way?

 

[Fredrik]

I'm not familiar with the notation "sec", but from what HallsofIvy wrote, I take it that sec θ is just 1/cos θ.

When I use this, and calculate dx=dx/dθ dθ, the integral simplifies to

\frac{1}{16}\int\frac{1}{cos^4\theta}d\theta

I don't know how to solve that, but maybe you do.

The thing that you're doing wrong is the derivative. There shouldn't be any logarithms in there.

 

[Pyrrhus]

Let's review what you got

2x = \sec\theta
\sqrt{4x^2-1}= tan\theta
dx = \frac{sec\theta tan\theta d\theta}{2}

so you will have:

\int \frac{(\frac{sec\theta}{2})^3}{tan\theta}\frac{sec\theta tan\theta d\theta}{2}

and working the terms:

\frac{1}{16} \int sec^4\theta d\theta

\frac{1}{16} \int sec^2\theta (1+tan^2\theta) d\theta

\frac{1}{16} \int sec^2\theta d\theta + sec^2\theta tan^2\theta d\theta

\frac{1}{16} [\int sec^2\theta d\theta + \int sec^2\theta tan^2\theta d\theta]

and you can finish it yourself, i believe...

 

[marlon]

use the fact that 1 + tan² = 1/cos² and write the integrandum as
(1/cos²(x) * 1/cos²(x))dx.

Then you also know that d(tan(x)) = dx/cos²(x)

So you get as integrandum 1/cos²(x)*d(tan(x)) and substitute the cos² by the above expression.

regards
marlon (sorry for the writing, i really should start using LaTex)

 

[COCoNuT]

Let's review what you got

2x = \sec\theta
\sqrt{4x^2-1}= tan\theta
dx = \frac{sec\theta tan\theta d\theta}{2}

so you will have:

\int \frac{(\frac{sec\theta}{2})^3}{tan\theta}\frac{sec\theta tan\theta d\theta}{2}

and working the terms:

\frac{1}{16} \int sec^4\theta d\theta

\frac{1}{16} \int sec^2\theta (1+tan^2\theta) d\theta

\frac{1}{16} \int sec^2\theta d\theta + sec^2\theta tan^2\theta d\theta

\frac{1}{16} [\int sec^2\theta d\theta + \int sec^2\theta tan^2\theta d\theta]

and you can finish it yourself, i believe...

\frac{1}{16} [\int sec^2\theta d\theta + \frac{1}{16}\int sec^2\theta tan^2\theta d\theta]

my work:
found the anti-derv.



\frac{1}{16}(tan \theta) + \frac{1}{16}(sec \theta)^2


ok now this is the part I am confused on... i know i need to draw a triangle and do something with it...
triangle is attached, but nothing is filled in. i took notes, but i don't know what to fill the values for the triangle.
im thinking that the values are...
a(Adjacent) = x^3
h = 1/2
O = \sqrt{(4x^2-1)}

how do you know what the a,h, and o sides are? i just looked at my notes and copied how it worked(i should take better notes).

ok now that i filled in the triangle, now to sub it back for \theta

\frac{1}{16}(tan \theta) + \frac{1}{16}(sec \theta)^2
will turn to...


\frac{1}{16}(\frac{\sqrt{(4x^2-1)}}{x^3}) + \frac{1}{16}(\frac{x^3}{2})^2

and that is my answer, is the correct?

 

[marlon]

The second integral should have solution tan³(x)/3, i think you made a mistake there.

 

[marlon]

[\int sec^2\theta tan^2\theta d\theta]

is equal to tan³(theta)/3

regards
marlon

 

[COCoNuT]

[\int sec^2\theta tan^2\theta d\theta]

is equal to tan³(theta)/3

regards
marlon

i don't see how you got that, i did the problem agian and got the same answer

 

[Pyrrhus]

Marlon is right, your first integral is correct but not your second.

Let's work out the second integral with substitution..

u=tan\theta
du=sec^2\theta d\theta

so we will have:
\int u^2 du

Integrating:
\frac{u^3}{3} + C

Substituting back:
\frac{tan^3\theta}{3} + C


your result should be:

\frac{tan\theta}{16} + \frac{tan^3\theta}{48} + C

Now to change it back:

Look at what we had above:
\sqrt{4x^2-1}= tan\theta

so
\frac{\sqrt{4x^2-1}}{16} + \frac{(\sqrt{4x^2-1})^3}{48} + C

 

[Pyrrhus]

I just noticed this is a definite integral... oh well just apply the theorem
and forget about the constant.

 

[COCoNuT]

sorry, you guys are right...


well anyway, with the correct results...

\frac{tan\theta}{16} + \frac{tan^3\theta}{48} + C


time to sub in theta...

\frac{\frac{\sqrt{(4x^2-1)}}{x^3}}{16} + (\frac{\frac{\sqrt{(4x^2-1)}}{x^3})^3}{48} + C

is that right? i drew a triangle and everything in my eariler post


Cyclovenom got a answer of \frac{\sqrt{4x^2-1}}{16} + \frac{(\sqrt{4x^2-1})^3}{48} + C

but i still have the x^3 in mines, what am i doing wrong? tan is o/a right? so i just subbed it in

 

[Pyrrhus]

On your triangle you should have on:

Hypotenuse: 2x
Opposite: \sqrt{4x^2-1}
Adjacent:1

 

[COCoNuT]

On your triangle you should have on:

Hypotenuse: 2x
Opposite: \sqrt{4x^2-1}
Adjacent:1

sorry to keep on bothering you guys, but how did you get those values?

 

[Pyrrhus]

Well, you started with 2x = sec\theta so if you read above HallsofIvy post on how to get a value that will work for the \sqrt{4x^2-1} which will be tan\theta. You can build a Triangle with that info.

You know sec\theta = \frac{Hypotenuse}{Adjacent} and you got
tan\theta = \frac{opposite}{adjacent}

so \frac{2x}{1} = \frac{Hypotenuse}{Adjacent} and \frac{\sqrt{4x^2-1}}{1} = \frac{opposite}{adjacent}

Also:

\frac{1}{cos\theta} = 2x

cos\theta = \frac{1}{2x}

and:

\frac{\sqrt{4x^2-1}}{1} = \frac{sin\theta}{cos\theta}

\frac{\sqrt{4x^2-1}}{1}\frac{1}{2x} = \frac{sin\theta}{cos\theta}cos\theta

sin\theta = \frac{\sqrt{4x^2-1}}{2x}

\frac{Opposite}{Hypotenuse} = \frac{\sqrt{4x^2-1}}{2x}

Do you see it?

 

[COCoNuT]

yea! thank you, i get it.

 

[COCoNuT]

Well, not "u= 2 sec θ" because there was no "u" in the integral!

But setting 2x= secθ will work very nicely (Notice where the 2 is!)

sin²θ+ cos²θ= 1 so, dividing both sides by cos²θ, tan²θ+ 1= sec²θ and then
sec²θ- 1= tan²θ.

That's the whole point of the substitution. If 2x= sec θ then 4x² =
sec² θ so 4x²- 1= tan²θ and you can drop the square root: \sqrt{4x^2-1}= \sqrt{tan^2\theta}= tan \theta.

(You might have been thinking of examples involving "x²- 4" where the substitution x= 2sec θ would give 4sec²- 4= 4(tan² θ)

lol, i just figured out what you tried to tell me, I am a bit slow. wow, that helped a lot. my teacher goes really fast, and doesn't explain the little details and the book is just weird. i already solved 6 problems on my own(14 more to go) and got them all right. thank you agian, i love this forum

 

[Pyrrhus]

Well, I'm glad i was of help
Keep solving problems!
and feel free to ask for help anytime as long as you will learn from it.