What is the Integral of dx/ (1+cos ^2(x)) Using Different Approaches?

In summary, the formula for calculating the integral of dx/ (1+cos ^2(x)) is ∫ dx/ (1+cos ^2(x)) = tan(x) + C, where C is the constant of integration. To solve this integral, the substitution method can be used by substituting u = cos(x) and du = -sin(x) dx. The domain and range of this integral are both all real numbers. It can also be used to find the area under a curve by taking the definite integral between two given points. Additionally, the integral can be solved using the trigonometric identity cos^2(x) = (1+cos(2x))/2.
  • #1
belleamie
24
0
for the life of me i can't seem to understand how to the the intergral of dx/ (1+cos ^2(x))?
 
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  • #2
Use a substitution

[tex] \tan\frac{x}{2}=t [/tex]

and some trigonometry.

Daniel.
 
  • #3
Daniel's approach probably works equally well; here's another approach:
[tex]\frac{1}{1+\cos^{2}x}=\frac{1}{\cos^{2}x}\frac{1}{1+\frac{1}{\cos^{2}x}}=(\frac{d}{dx}tan(x))\frac{1}{2+\tan^{2}x}[/tex]
Thus, setting [tex]u=tan(x)[/tex], we have [tex]\frac{du}{dx}dx=du[/tex], that is:
[tex]\int\frac{dx}{1+\cos^{2}x}=\int\frac{du}{2+u^{2}}[/tex]
 

Related to What is the Integral of dx/ (1+cos ^2(x)) Using Different Approaches?

1. What is the formula for calculating the integral of dx/ (1+cos ^2(x))?

The formula for calculating the integral of dx/ (1+cos ^2(x)) is ∫ dx/ (1+cos ^2(x)) = tan(x) + C, where C is the constant of integration.

2. How do you solve the integral of dx/ (1+cos ^2(x))?

To solve the integral of dx/ (1+cos ^2(x)), you can use the substitution method by substituting u = cos(x) and du = -sin(x) dx. This will transform the integral into ∫ du/ (1+u^2), which can be solved using the arctan formula.

3. What is the domain and range of the integral of dx/ (1+cos ^2(x))?

The domain of the integral of dx/ (1+cos ^2(x)) is all real numbers, while the range is also all real numbers.

4. Can the integral of dx/ (1+cos ^2(x)) be used to find the area under a curve?

Yes, the integral of dx/ (1+cos ^2(x)) can be used to find the area under a curve. It represents the antiderivative of the function, and the definite integral can be used to find the area under the curve between two given points.

5. Are there any other methods to solve the integral of dx/ (1+cos ^2(x))?

Yes, apart from the substitution method, the integral of dx/ (1+cos ^2(x)) can also be solved using the trigonometric identity cos^2(x) = (1+cos(2x))/2. This will transform the integral into a simpler form that can be solved using basic integration techniques.

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