Struggling with Trig Integrals? Get Help Here!

In summary, the conversation discusses how to solve the integral \int\frac{cosxdx}{\sqrt{1+cosx}} using various methods, including substitution and trigonometric identities. The final solution involves rewriting the integrand as \frac{cos^2(x/2)-sin^2(x/2)}{\sqrt{2}cos(x/2)} and using the half angle and double angle formulas.
  • #1
noblerare
50
0

Homework Statement



[tex]\int[/tex][tex]\frac{cosxdx}{\sqrt{1+cosx}}[/tex]

Homework Equations



n/a

The Attempt at a Solution



Well I tried:
u=cosx
x=arccosx
dx=[tex]\frac{-du}{\sqrt{1-u^2}}[/tex]

Plugging them back into the integral gives me:

-[tex]\int[/tex][tex]\frac{udu}{\sqrt{1-u^2}\sqrt{1+u}}[/tex]

I don't know where to go from there.

I've also tried:

[tex]\int[/tex][tex]\frac{cosx\sqrt{1-cosx}dx}{sin^2x}[/tex]

or:

[tex]\int[/tex][tex]\sqrt{\frac{cos^2x}{1+cosx}}dx[/tex]

But I don't know where to go from any of the above attempts. Any help would be greatly appreciated.
 
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  • #2
Ok ... now I need a hint, lol.
 
  • #3
I figured it out by mere chance. Note that [tex]\frac{1}{\sqrt{1+cos(x)}}=\frac{1}{\sqrt{2}cos(x/2)}[/tex] and that [tex]cos(x)=cos^2(x/2)-sin^2(x/2)[/tex]. I used the half angle and double angle formulas, respectively. Wikipedia had the answer to [tex]\int sin(ax)tan(ax)dx[/tex] ( http://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functions )
 
Last edited:
  • #4
wow cool! I'm glad to see that you figured it out, and I understand how you got the two equations you wrote but I don't see what I should do to solve the problem.
 
  • #5
So I'll have:

[tex]\int[/tex][tex]\frac{cos^2(x/2)-sin^2(x/2)}{\sqrt{2}cos(x/2)}[/tex]

Which I simplify to:

[tex]\frac{1}{\sqrt{2}}[/tex][tex]\int[/tex]cos(x/2)dx - [tex]\frac{1}{\sqrt{2}}[/tex][tex]\int[/tex][tex]\frac{sin^2(x/2)}{cos(x/2)}[/tex]

Is that what I'm supposed to do?

Where did the sin(ax)tan(ax)dx thing come from?
 
  • #6
So the left integral is easily solvable, but the right integral needs a little work, and I just pointed out that you can rewrite the right side's terms as the product of a sine and tangent.
 
  • #7
ah...i see, wow thanks, jhicks!
 

1. What is a trigonometric integral?

A trigonometric integral is an integral that involves trigonometric functions (such as sine, cosine, tangent, etc.) in the integrand. It can be solved using various integration techniques, such as substitution, integration by parts, and special trigonometric identities.

2. How do I integrate a trigonometric function?

To integrate a trigonometric function, you can use various integration techniques depending on the form of the integrand. Some common techniques include substitution, integration by parts, and using special trigonometric identities. It is important to have a good understanding of these techniques and practice solving integrals to become proficient at integrating trigonometric functions.

3. What are some common trigonometric identities that can be used to solve integrals?

Some common trigonometric identities that can be used to solve integrals include the Pythagorean identities, double angle identities, and half angle identities. These identities can help simplify the integrand and make it easier to integrate using other techniques.

4. Can I use a calculator to evaluate trigonometric integrals?

Yes, some calculators have built-in functions for evaluating trigonometric integrals. However, it is important to note that these functions may not work for all types of integrals, and it is still important to understand the underlying concepts and techniques for solving integrals.

5. Are there any tips for solving tricky trigonometric integrals?

One tip for solving tricky trigonometric integrals is to try manipulating the integrand using trigonometric identities or algebraic techniques. It can also be helpful to rewrite the integral in terms of a different trigonometric function to make it easier to integrate. Practice and familiarity with different integration techniques can also help when faced with tricky integrals.

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