Difficult Integral of trig and sqrt functions

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SUMMARY

The integral ∫ (sinx)^2 / √(a - b*cos(x)) dx, where a and b are constants, presents significant challenges when using standard calculus techniques such as substitution and the product rule. Users have reported difficulties in solving this integral, often leading to dead ends. A recommended approach is to utilize the software Maple, which has a built-in solver capable of handling non-standard integrals and provides solutions in terms of Elliptic functions. For specific values, such as a = 4 and b = 2, Maple successfully computes the integral, demonstrating its effectiveness for complex integrals.

PREREQUISITES
  • Understanding of integral calculus
  • Familiarity with trigonometric functions
  • Knowledge of Elliptic functions
  • Experience using Maple software for mathematical computations
NEXT STEPS
  • Explore the capabilities of Maple for solving complex integrals
  • Study Elliptic functions and their applications in integration
  • Learn advanced integration techniques beyond standard calculus methods
  • Investigate other mathematical software options for integral solving, such as Mathematica or MATLAB
USEFUL FOR

Mathematicians, engineering students, and anyone dealing with complex integrals in their studies or research will benefit from this discussion.

dm164
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I'm having troubles solving this integral.

∫ (sinx)^2 / √( a - b*cos(x)) dx ; a & b just constants

I've tried many routes but get stuck.
And when substitution and product rule methods seem to fail. Are there other methods useful to go to when integrating strange functions?
 
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dm164 said:
I'm having troubles solving this integral.

∫ (sinx)^2 / √( a - b*cos(x)) dx ; a & b just constants

I've tried many routes but get stuck.
And when substitution and product rule methods seem to fail. Are there other methods useful to go to when integrating strange functions?

When standard methods fail, which is often for "strange" functions I usually try to see if Maple gives an answer. Maple seems to have a built in solver that knows many of the non-standard and lesser studied integrals. For this example I set a = 4 and b = 2 just to see what would happen and Maple gives an answer in terms of Elliptic functions. You aren't going to solve it with the standard calculus methods.
 

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