Integrating cos(x)/1+sin^2(x)dx - Methods & Tips

  • Thread starter mickellowery
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In summary, the conversation suggests using a substitution method to integrate the given expression, specifically by setting u = sin(x) and rewriting the integral in the form \int \frac{du}{1+u^2}.
  • #1
mickellowery
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What method would I use to integrate cos(x)/1+sin^2(x)dx? I was thinking by parts but it isn't working for me.
 
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  • #2
mickellowery said:
What method would I use to integrate cos(x)/1+sin^2(x)dx? I was thinking by parts but it isn't working for me.
Try a u substitution.
 
  • #3
mickellowery said:
What method would I use to integrate cos(x)/1+sin^2(x)dx? I was thinking by parts but it isn't working for me.

Try a substitution. Notice that cos(x) is the derivative of sin(x).
 
  • #4
Wouldn't the derivative of sin^2(x) be 2sin(x)cos(x)? I think I'm still a little lost.
 
  • #5
Try getting it into this format

[tex] \int \frac{du}{1+u^2}[/tex]
 

1. What is the integration method for cos(x)/1+sin^2(x)dx?

The integration method for cos(x)/1+sin^2(x)dx is to use the substitution method. Let u = sin(x) and du = cos(x)dx, then the integral becomes ∫du/(1+u^2), which can be solved using partial fractions or by using the inverse tangent function.

2. Can the integral be solved using trigonometric identities?

Yes, the integral cos(x)/1+sin^2(x)dx can also be solved using trigonometric identities, specifically tan^2(x) + 1 = sec^2(x). This will result in an integral of the form ∫sec^2(x)/(1+tan^2(x)), which can be solved using the substitution method mentioned above.

3. Are there any other methods to solve this integral?

Yes, apart from the substitution and trigonometric identities method, this integral can also be solved using the Weierstrass substitution. Let u = tan(x/2), then the integral becomes ∫2du/(u^2+1). This can be solved using partial fractions or by using the inverse hyperbolic tangent function.

4. Is there a shortcut or trick to solving this integral?

No, there is no shortcut or trick to solving this integral. It requires knowledge of various integration techniques and the ability to recognize the appropriate method to use in each case.

5. Can this integral be solved using software or calculators?

Yes, this integral can be solved using various software or online calculators that have the ability to perform symbolic integration. However, it is important to have a basic understanding of the integration methods involved in order to verify the accuracy of the result.

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