Can the Integral of 1/(1 + cosx)^2 be Simplified Further with Substitutions?

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In summary, the formula for the integral of 1/(1 + cosx)^2 is 2tan(x/2) + C. It is derived by using the substitution u = tan(x/2) and the identity 1 + tan^2(x/2) = sec^2(x/2). The limits of integration can be any values of x that result in a valid value for tan(x/2). Other methods such as the Weierstrass substitution or the half-angle formula for cosine can also be used to solve this integral, but the substitution method is the most common. This integral has applications in physics and engineering, particularly in problems involving vibrations and oscillations, as well as in the calculation of geometric shapes like
  • #1
redshift
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Just finished working out the integral of 1/(1 + cosx)^2 as
(1/2)(tan(x/2)) + (1/6)(tan (x/2))^3 + C
I'm just wondering whether it's possible to use any substitutions to simplify this further.

regards
 
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  • #2
I think it's simple enough..
 
  • #3
hi.
i've jux come across calculating the exact same integral (except that it's definite!). i tried to solve it using different substitutions bt all in vain. So. culd somebody post the complete solution here.
 

1. What is the formula for the integral of 1/(1 + cosx)^2?

The formula for the integral of 1/(1 + cosx)^2 is 2tan(x/2) + C.

2. How is the formula derived?

The formula is derived by using the substitution u = tan(x/2) and the identity 1 + tan^2(x/2) = sec^2(x/2). This leads to the integral becoming 2/(1 + u^2), which can be solved using the inverse tangent function and the integration by substitution rule.

3. What are the limits of integration for this integral?

The limits of integration for this integral can be any values of x that result in a valid value for tan(x/2). This means that the limits can be any values between -π and π, excluding odd multiples of π/2.

4. Can the integral of 1/(1 + cosx)^2 be solved using other methods?

Yes, the integral can also be solved using the Weierstrass substitution or by using the half-angle formula for cosine. However, the substitution method is the most common and straightforward approach.

5. What are the applications of this integral?

This integral is often used in physics and engineering, specifically in problems involving vibrations and oscillations. It is also used in the calculation of certain geometric shapes, such as the area of an ellipse.

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