Trigonometric Substitution Problem

  • Thread starter Lancelot59
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  • #1
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This problem looks relatively simple, but the coefficient in front of the variable is causing issues:

[tex]\int{\sqrt{1-4x^{2}}}dx[/tex]

So I started like this:

[tex]x=sin(\theta)[/tex]
[tex]dx=cos(\theta)d\theta[/tex]
[tex]\int{\sqrt{1-4sin^{2}(\theta)}cos(\theta)d\theta}[/tex]

Normally you can remove the constant from the root and go about the rest of the integral, but I'm stumped on this. How can I break this down into something more manageable? I've tried several identities but nothing has worked yet.

A pointer in the right direction would be great. Thanks in advance!
 

Answers and Replies

  • #2
35,045
6,782
Use the substitution sin(theta) = 2x. Rather than trying to remember which substitution to use, I draw a right triangle, labellings its sides according to the quantities in the radical. My triangle has a hypotenuse of 1, opposite side of 2x, and adjacent side of sqrt(1 - 4x^2).
 
  • #3
634
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So you just treat the root as the Pythagorean theorem? Makes sense.

To make sure I understand this fully, if my equation was [tex]\int{\sqrt{1+16x^{2}}}dx[/tex]

I would say that one of triangle's sides was 1, and the other was 4x? Making the hypotenuse the square root of 1-16x^2?

Okay, so going back to the original I can say that sin(theta) = 2x/1, and then isolate x to get x=sin(theta)/2. Putting that in it squares and cancels out the 4. Wonderful, thanks for the help!
 
  • #4
35,045
6,782
So you just treat the root as the Pythagorean theorem? Makes sense.

To make sure I understand this fully, if my equation was [tex]\int{\sqrt{1+16x^{2}}}dx[/tex]

I would say that one of triangle's sides was 1, and the other was 4x?
Yes.
Making the hypotenuse the square root of 1-16x^2?
No, the hypotenuse would be sqrt(1 + 16x^2).
Okay, so going back to the original I can say that sin(theta) = 2x/1, and then isolate x to get x=sin(theta)/2. Putting that in it squares and cancels out the 4. Wonderful, thanks for the help!
 
  • #5
634
1
I hit the minus sign by accident. :p
 
  • #6
35,045
6,782
I hope you don't get a job at the bank I go to :yuck:

(Just kidding!)
 
  • #7
634
1
I hope you don't get a job at the bank I go to :yuck:

(Just kidding!)

Eww...banking.:yuck:
 

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