- #1
Inquisitive_Mind said:Use t method? (i.e. set t=tan(x/2), such that cos(x)=(1-t^2)/(1+t^2))
dextercioby said:I told u it doesn't work...
[tex] \int \frac{d\theta}{(3-2\cos\theta)^{2}} [/tex]
,via the substitution
[tex] \cos\theta\rightarrow x [/tex]
becomes
[tex] -\int \frac{dx}{\sqrt{1-x^{2}}(3-2x)^{2}} [/tex]
which is very irrational.
So your
[tex] \frac{A}{(3-2x)^{2}}+\frac{B}{\sqrt{1-x}}+\frac{C}{\sqrt{1+x}} [/tex]
doesn't lead anywhere.
Daniel.
dextercioby said:I knew the decomposition was wrong,i told him that substitution would lead nowhere.
Anyways,my result is
[tex] \int \frac{d\theta}{(3-2\cos\theta)^{2}}=\frac{4}{5}\frac{\tan\frac{\theta}{2}}{1+5\tan^{2}\frac{\theta}{2}}+\frac{6\sqrt{5}}{25}\arctan(\sqrt{5}\tan\frac{\theta}{2})+C [/tex]
Daniel.
A rational function integral is the process of finding the antiderivative of a rational function, which is a function that can be expressed as the quotient of two polynomials. It is a type of indefinite integral and is often used in calculus to solve problems related to areas and volumes.
To solve a rational function integral, you can use a variety of techniques such as substitution, integration by parts, or partial fractions. The specific method used will depend on the complexity of the rational function and the tools available. It is important to follow the rules of integration and to simplify the expression as much as possible before attempting to integrate.
Yes, there are a few special cases when integrating rational functions. One is when the degree of the numerator is greater than or equal to the degree of the denominator, in which case the integral will involve a polynomial division. Another case is when the rational function contains a quadratic expression in the denominator, which may require the use of trigonometric substitutions. Additionally, if the rational function has any terms involving square roots, it may be necessary to use a u-substitution.
Rational function integrals have many real-life applications, particularly in the field of physics. They can be used to calculate the work done by a variable force, the center of mass of an object, and the volume of a solid of revolution. In engineering, rational function integrals are used to find the moment of inertia of an object and the deflection of a beam under load. They are also useful in economics for calculating marginal cost and marginal revenue.
Yes, there are a few common mistakes that should be avoided when integrating rational functions. These include forgetting to use the proper rules of integration, not simplifying the expression before integrating, and incorrectly setting up the integral by forgetting to include all terms. It is also important to check for any potential discontinuities or vertical asymptotes in the rational function before integrating. Additionally, it is always a good idea to double-check your answer by differentiating it to ensure it is correct.