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Alexx1
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Can someone help me with this integral?
Integral: 1/(1+cos(x)+sin(x))
Integral: 1/(1+cos(x)+sin(x))
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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.
D H said: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!
Good so far.Alexx1 said:Than I get
ln (t+1)
= ln (tan(x/2) +1)
Whoa! What's this last step?= ln ((sin(x/2) / cos(x/2)) +1)
= ln (sin(x/2) +1) - ln(cos(x/2)+1)
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.
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.
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.
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.
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.
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.