- #1
RobertT
- 26
- 0
[tex]\int \frac{\cos^2 x}{(1+\epsilon\cos x)^3}\,dx[/tex]
Where, [tex]\epsilon > 0[/tex] is a real number constant.
Where, [tex]\epsilon > 0[/tex] is a real number constant.
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Nebuchadnezza said:My calculator gives me the answer as:
[tex]\int {\frac{{\cos {{\left( x \right)}^2}}}{{{{\left( {1 + n\cos \left( x \right)} \right)}^2}}}dx} = \frac{1}{2}\left( {\frac{{\sin \left( x \right)\left( { - 4{n^2}\cos \left( x \right) - 3n + \cos \left( x \right)} \right)}}{{{{\left( {{n^2} - 1} \right)}^2}{{\left( {n\cos \left( x \right) + 1} \right)}^2}}} + \frac{{2\left( {2{n^2} + 1} \right)\tanh \left( {\frac{{\left( {n - 1} \right)\tan \left( {\frac{x}{2}} \right)}}{{\sqrt {{n^2} - 1} }}} \right)}}{{{{\left( {{n^2} - 1} \right)}^{\frac{5}{2}}}}}} \right) + C[/tex]
Perhaps some fancy trig substitution will do the trick along with some partial fractions =)
Nebuchadnezza said:A small typo from my side, but the answer is still correct.
And here is my "Magical" calculator. Maple 13 gave me a tad uglier answer so I decided to use this one instead:
http://www.wolframalpha.com/input/?i=integrate+cos%28x%29^2%2F%281%2Bn*cos%28x%29%29^3
Just out of curiosity, what is this integral for?
Trigonometric integration is the process of finding the integral of a function that contains one or more trigonometric terms. It involves using various trigonometric identities and techniques to simplify the integral and find a solution.
Trigonometric integration can be challenging because it requires a thorough understanding of trigonometric identities and techniques, as well as the ability to recognize and apply them correctly. It also often involves multiple steps and can require a lot of algebraic manipulation.
Some tips for solving trigonometric integrals include using trigonometric identities and substitution, factoring out common terms, and simplifying the integral before attempting to solve it. It can also be helpful to practice and become familiar with common trigonometric integrals.
One way to check if your solution is correct is to differentiate it and see if you get back to the original function. You can also use online integration calculators or ask for feedback from a math tutor or professor.
Trigonometric integration has many real-world applications, such as in physics, engineering, and economics. It can be used to model and solve problems involving waves, oscillations, and periodic phenomena. It can also be used to calculate areas, volumes, and other quantities in real-world situations.