# Figuring out the solution of the following Integral

## Homework Statement

Hello,I'm having problems figuring out the solution of the following Integral.Any hints would be much appreciated!

There it is:

I= sin(x)/sqrt{4sin(x)^2+cos(x)^2} dx

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$$I=\int\frac{sin(x) dx}{\sqrt{4sin^2(x)+cos^2(x)}}$$

O.O wow thats...a toughie, no substitution seems to work...you could try tan(x/2)=y but you'd get a polynomial of degree 4 on the square root...

It would be so much nicer if that 4 was gone...

ok you could try getting rid of those sin^2 x by using those formulas
$$sin^2 (x) = \frac{1-cos 2x}{2}$$
$$cos^2 (x) = \frac{1+cos 2x}{2}$$

that might lead you somewhere decent but you'll have some problems with sin(x) on the top but square root of acos2x + or - something...

Yeah,I've tried everything you've listed above,but had no success lol.
Anyway,thank you !

Gib Z
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Not expressible in elementary functions or in a closed form.

dynamicsolo
Homework Helper

## Homework Statement

Hello,I'm having problems figuring out the solution of the following Integral.Any hints would be much appreciated!

There it is:

I= sin(x)/sqrt{4sin(x)^2+cos(x)^2} dx

It's always a bad sign when it doesn't show up in Gradshteyn & Ryzhik...

Are you sure that you're asked for the indefinite integral? The closest entry I found was [3.676 #1] in G&R, 5th edition:

rewrite $$4sin^2(x)+cos^2(x)$$ as $$1+3sin^2(x)$$ and use

definite integral from 0 to pi/2 of $$\int\frac{sin(x) dx}{\sqrt{1+p^2sin^2(x)}}$$

= (1/p) arctan p , with p^2 = 3 .

It looks like this may be one of those integrals that only has nice solutions for *certain* definite limits, but not in general...

so yea I tried it and my integral has complex numbers, so i'm going to go recheck my work.

HallsofIvy
Homework Helper
$$I=\int\frac{sin(x) dx}{\sqrt{4sin^2(x)+cos^2(x)}}= \int\frac{sin(x)dx}{\sqrt{4- 5cos^2(x)}}$$

Let u= cos(x) so du= sin(x)dx and the integral becomes
$$\int\frac{du}{\sqrt{4- 5u^2}}= \frac{1}{2}\int\frac{du}{\sqrt{1- \frac{5}{4}u^2}}$$
Letting $sin(\theta)= \sqrt{5}/2 u$ should convert it to a rational integral.

Gib Z
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Thats some really nice work there Halls!! I wish i'd seen that =] Just One tiny mistake though, u=cos x so du = - sin x dx, so the actual integral is the one you gave, just with a negative sign stuck out the front.

dynamicsolo
Homework Helper
$$I=\int\frac{sin(x) dx}{\sqrt{4sin^2(x)+cos^2(x)}}= \int\frac{sin(x)dx}{\sqrt{4- 5cos^2(x)}}$$

I am likewise chagrined to have missed that; I was focusing on the sine-squared term, rather than the cosine-squared one.

One other little thing, though: 4 (sin x)^2 + (cos x)^2 =
4 [1 - (cos x)^2] + (cos x)^2 =
4 - 4 (cos x)^2 + (cos x)^ 2 =
4 - 3 (cos x)^2 , no?

HallsofIvy