How Can I Integrate 1/(sin(x)+a) dx?

In summary, the problem is to integrate ∫dx/(sin(x)+a), where a is a constant. The attempt to multiply the top and bottom by (sin(x)-a) seemed promising, but the second term (-∫a/(sin2(x)-a2)dx) caused some difficulty. The solution involves rewriting the integrand in terms of tan(x/2) and using the substitution tan(x/2)=t.
  • #1
msbell1
25
0

Homework Statement


Integrate ∫dx/(sin(x)+a), where a is a constant.

Homework Equations


The Attempt at a Solution


I have been working on this for a while, and for some reason I can't figure it out. The attempt that seemed the most promising to me was to multiply top and bottom by (sin(x)-a), which gave
∫(sin(x)-a)/(sin2(x)-a2) dx.
I could integrate the first term (∫sin(x)/(sin2(x)-a2)dx) by substituting 1-cos2(x) for sin2x, and then using partial fractions. However, the second term (-∫a/(sin2(x)-a2)dx) is causing me some trouble. Actually, with the second term, I again used partial fractions, but then I end up with (sin(x) +/- a) in the denominator, which ends up looking about the same as what I started with. Is there a substitution that would make this problem simple? Thanks.
 
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  • #2
The integrand is a rational function of sin(x): rewrite it in terms of tan(x/2) and use the substitution tan(x/2)=t.
(sin(x)=2tan(x/2)/(1+tan2(x/2), cos(x)=(1-tan2(x/2)/(1+tan2(x/2). )

ehild
 

1. What is the general formula for integrating 1/(sin(x)+a)?

The general formula for integrating 1/(sin(x)+a) is -1/a * ln|cos(x)+(1/a)sin(x)| + C where C is the constant of integration.

2. Can you provide an example of how to integrate 1/(sin(x)+a)?

Let's say we want to integrate 1/(sin(x)+2). Using the general formula, we have -1/2 * ln|cos(x)+2sin(x)| + C as our final answer.

3. Is there a specific method or technique for integrating 1/(sin(x)+a)?

Yes, there is a specific technique called substitution that is commonly used for integrating expressions with trigonometric functions. In this case, we would let u = sin(x)+a and then solve for dx to substitute into the integral.

4. Are there any restrictions or limitations when integrating 1/(sin(x)+a)?

Yes, there are some restrictions to be aware of. This integral is only valid when a ≠ 0 and when the denominator sin(x)+a does not equal 1. If these conditions are not met, the integral will not converge.

5. Can you explain the significance of the constant of integration in this integral?

The constant of integration is a crucial component in the integration process. It represents the family of curves that all have the same derivative, in this case being 1/(sin(x)+a). It allows us to account for all possible solutions to the integral.

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