# Integral: 1/(1+cos(x)+sin(x)) help

• Alexx1
In summary, The conversation is about solving an integral involving 1/(1+cos(x)+sin(x)). The process involves using trig substitutions, particularly the half angle formula for tan(x/2). The final answer is ln(tan(x/2)+1), which is achieved by first finding the correct substitutions and then simplifying the expression. The person asking for help had initially made a mistake, but was able to correct it and reach the correct answer.
Alexx1
Can someone help me with this integral?

Integral: 1/(1+cos(x)+sin(x))

Last edited:

Solving this will take some adroit trig substitutions. Half angle formulae, particularly that for tan(x/2), will come in handy here.

D H said:
Solving this will take some adroit trig substitutions. Half angle formulae, particularly that for tan(x/2), will come in handy here.

I've tried it, but my answer isn't correct..

cos(x) = (1 - (tan(x/2))^2) / (1 + (tan(x/2))^2)

sin(x) = (2*tan(x/2)) / (1 + (tan(x/2))^2)

And t = tan(x/2) ==> x = 2 bgtan(t) ==> dx = 2/(1+x^2)

If I do it like that i become this integral:

1/(t+1) dt

But that's not correct. Have I done something wrong?

That is exactly right, but you haven't finished yet. Continue on!

D H said:
That is exactly right, but you haven't finished yet. Continue on!

Than I get

ln (t+1)

= ln (tan(x/2) +1)

= ln ((sin(x/2) / cos(x/2)) +1)

= ln (sin(x/2) +1) - ln(cos(x/2)+1)

And what now?

D H said:
That is exactly right, but you haven't finished yet. Continue on!

Than I get

ln (t+1)

= ln (tan(x/2) +1)

= ln ((sin(x/2) / cos(x/2)) +1)

= ln (sin(x/2) +1) - ln(cos(x/2)+1)

And what now?

Alexx1 said:
Than I get

ln (t+1)

= ln (tan(x/2) +1)
Good so far.

= ln ((sin(x/2) / cos(x/2)) +1)

= ln (sin(x/2) +1) - ln(cos(x/2)+1)
Whoa! What's this last step?

For that matter, why do you need to go beyond ln(tan(x/2)+1) ? That is a perfectly good answer in and of itself.

D H said:
Good so far.

Whoa! What's this last step?

For that matter, why do you need to go beyond ln(tan(x/2)+1) ? That is a perfectly good answer in and of itself.

Ow, I'm verry sorry.. I made a mistake, I read the wrong answer in my book.. ln (tan(x/2) +1) is the correct answer..

## 1. What is the purpose of using the integral 1/(1+cos(x)+sin(x))?

The integral 1/(1+cos(x)+sin(x)) is commonly used in mathematics and physics to solve problems related to periodic functions. It allows for the calculation of the area under the curve of a function that oscillates between positive and negative values.

## 2. How do I solve the integral 1/(1+cos(x)+sin(x))?

To solve this integral, you can use a variety of techniques such as substitution, integration by parts, or trigonometric identities. It is recommended to use multiple methods to check your work and ensure accuracy.

## 3. Can the integral 1/(1+cos(x)+sin(x)) be solved analytically?

Yes, this integral can be solved analytically using the methods mentioned above. However, the result may be expressed in terms of special functions such as the inverse tangent function or hyperbolic functions.

## 4. How is the integral 1/(1+cos(x)+sin(x)) related to the Fourier series?

The Fourier series is a mathematical tool used to represent periodic functions as a sum of sine and cosine functions. The integral 1/(1+cos(x)+sin(x)) is closely related to the Fourier series as it can be used to determine the coefficients of the series.

## 5. Are there any real-world applications of the integral 1/(1+cos(x)+sin(x))?

Yes, this integral has many real-world applications in fields such as engineering, physics, and economics. It can be used to model and solve problems related to oscillating systems, such as the motion of a pendulum or the behavior of alternating electric currents.

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