# Trigonometric Substitution Problem

This problem looks relatively simple, but the coefficient in front of the variable is causing issues:

$$\int{\sqrt{1-4x^{2}}}dx$$

So I started like this:

$$x=sin(\theta)$$
$$dx=cos(\theta)d\theta$$
$$\int{\sqrt{1-4sin^{2}(\theta)}cos(\theta)d\theta}$$

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

Mark44
Mentor
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).

So you just treat the root as the Pythagorean theorem? Makes sense.

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

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!

Mark44
Mentor
So you just treat the root as the Pythagorean theorem? Makes sense.

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

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!

I hit the minus sign by accident. :p

Mark44
Mentor
I hope you don't get a job at the bank I go to :yuck:

(Just kidding!)

I hope you don't get a job at the bank I go to :yuck:

(Just kidding!)

Eww...banking.:yuck: