Evaluate an Integral - No Clue

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Homework Help Overview

The discussion revolves around evaluating the integral \(\int \frac{x}{1 + \sin(x)} \, dx\), which falls under the subject area of calculus, specifically integral calculus.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants express uncertainty about the appropriate rules and approaches to use for the integral. Some suggest considering whether it is a definite integral, while others propose using a power series approximation. There are also suggestions to manipulate the integral by multiplying by a specific factor and separating terms to facilitate integration by parts.

Discussion Status

The discussion is active, with participants sharing various approaches and suggestions. Some guidance has been offered regarding potential methods, such as integration by parts and trigonometric identities, but there is no explicit consensus on the best path forward.

Contextual Notes

Participants note the complexity of the integral and express concerns about the difficulty of finding a solution. There is also a mention of the integral being indefinite, which may affect the approach taken.

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Evaluate an Integral - No Clue!

Homework Statement



[tex]\int[/tex] x/(1 + sin(x)) dx

Homework Equations





The Attempt at a Solution



I don't know what rules to use or what approach to take! Please help!
 
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is this a definite integral? that may simplify it, otherwise i think its pretty complex... can you use a power series approximation?
 


No indefinite and yeah looks complex to me...
 


I would start out by multiplying by (1-sin(x))/(1-sin(x)).
 


nice suggestion Dick
 


lanedance said:
nice suggestion Dick

Thanks! I'm not sure how easily it goes from there. Looks like some integration by parts.
 


So I got it to x/(1 + sin(x)) = x(1 - sin(x))/(cos2(x)) but now I am stuck again... D:
 


If you separate (1-sin(x)/cos2(x)) you can do the following...
x(1/cos2(x) - sin(x)/cos2(x)) =
x(sec2(x)-tan(x)sec(x)) =
∫xsec2(x) - ∫xtan(x)sec(x)

Now you can use integration by parts on both integrals?
 
Last edited:


hint:recall from elementary trigonomtry that
1+sin(x)=2 sin(x/2+pi/4)^2
 

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