Solving Integrals with Constants: Tips and Techniques for Success

  • Thread starter LagrangeEuler
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In summary, the student is trying to find the integral of a function which may be defined for certain values of y in (0,pi)+k*pi for integer values of k, and may be imaginary for certain choices of constants.
  • #1
LagrangeEuler
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Homework Statement


How to solve integral
[tex]\int \frac{dy}{\sqrt{C_1-K \cos y-\frac{C_2}{\sin^2 y}}}[/tex]

Homework Equations


##C_1,C_2## and ##K## are constants.

The Attempt at a Solution


I am not sure which method I should use here or is this integral maybe eliptic? Please give me the hint. Which supstitution or method and I will solve integral to the end.[/B]
 
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  • #2
Start by eliminating the fractions under the square root.
 
  • #3
micromass said:
Start by eliminating the fractions under the square root.
Ok
[tex]C_1-K\cos y-\frac{C_2}{\sin^2 y}=\frac{C_1 \sin^2 y-K\cos y\sin^2 y-C_2}{\sin^2 y}[/tex]
So now I have
[tex]\int \frac{\sin ydy}{\sqrt{C_1 \sin^2 y-K\cos y\sin^2 y-C_2}} [/tex]
 
Last edited:
  • #4
LagrangeEuler said:
Ok
[tex]C_1-K\cos y-\frac{C_2}{\sin^2 y}=\frac{C_1 \sin^2 y-K\cos y\sin^2 y-C_2}{\sin^2 y}[/tex]
So now I have
[tex]\int \frac{\sin ydy}{\sqrt{C_1 \sin^2 y-K\cos y\sin^2 y-C_2}} [/tex]
If I take
##\cos y =t##
then
##-\sin ydy=dt##
[tex]I=-\int\frac{dt}{\sqrt{C_1(1-t^2)-Kt(1-t^2)-C_2}}[/tex]
[tex]I=-\int \frac{dt}{\sqrt{Kt^3-C_1t^2-Kt+C_1-C_2}} [/tex]
 
  • #5
If your polynomial at the denominator were of degree 2, you would write it under its canonical form, and depending upon the coefficients and the discriminant, you would use a substitution of type Arcsin, Argsinh, or Argcosh to get rid of the square root. I don't know if you can do the same thing for polynomials of degree 3.
 
  • #6
LagrangeEuler said:
If I take
##\cos y =t##
then
##-\sin ydy=dt##
[tex]I=-\int\frac{dt}{\sqrt{C_1(1-t^2)-Kt(1-t^2)-C_2}}[/tex]
[tex]I=-\int \frac{dt}{\sqrt{Kt^3-C_1t^2-Kt+C_1-C_2}} [/tex]

You can express the integral in terms of Elliptic functions but it is very messy.
 
  • #7
Are there limits on your integration?
It looks like the function of y might be defined for y in (0,pi)+k*pi for integer values of k, and may be imaginary for certain choices of constants.
 

What is an integral?

An integral is a mathematical concept that represents the accumulation of a quantity over a certain interval. It can be thought of as the inverse operation of differentiation, and is used to find the area under a curve or to solve problems involving rates of change.

What is the process for solving an integral?

The process for solving an integral involves finding the antiderivative of the function, which is the original function that, when differentiated, gives the integrand. This is typically done using integration techniques such as substitution, integration by parts, or partial fractions.

What is the difference between definite and indefinite integrals?

A definite integral has specific limits of integration and represents a numerical value, while an indefinite integral does not have limits and represents a general function. In other words, a definite integral gives a specific answer, while an indefinite integral gives a family of solutions.

How do you determine the limits of integration for a definite integral?

The limits of integration for a definite integral can be determined by looking at the problem and identifying the start and end points of the interval over which the quantity is being accumulated. These limits will then be plugged into the integral to calculate the final numerical value.

What are some common techniques for solving integrals?

Some common techniques for solving integrals include substitution, integration by parts, partial fractions, trigonometric substitution, and using tables of integrals. It is important to choose the appropriate technique based on the form of the integrand and the given problem.

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