Integration of x^2/(xsinx+cosx)^2

  • Thread starter JasonHathaway
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In summary, the conversation is about a calculus problem involving integration and trigonometric substitutions, specifically the problem of finding the integral of x^2 / (xsinx + cosx)^2. The person has already completed their calculus course but is still curious about the problem. They mention using integration by parts or trigonometric substitutions to solve it. One person asks for clarification on how the solution involves sec(x), and another person explains that it is because sec(x) = 1/cos(x).
  • #1
JasonHathaway
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Hi everyone,

First of all, this isn't really a "homework", I've completed my calculus course and I'm just curious about this problem.

Homework Statement



[itex]\int\frac{x^{2}}{(xsinx+cosx)^{2}} dx[/itex]

Homework Equations



Trigonometric substitutions, integration by parts maybe?

The Attempt at a Solution



This is a solved problem.

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How does [itex]\int\frac{x^{2}}{(xsinx+cosx)^{2}} dx[/itex] become [itex]\int xsecx \frac{xcosx}{(xsinx+cosx)^{2}} dx[/itex]?
 
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  • #2
Just because [tex]sec(x)=\frac{1}{cos(x)}[/tex]
 
  • #3
Did it multiply the numerator and denominator by [itex]\frac{cosx}{cosx}[/itex], which is [itex]cosx secx[/itex], and then both of [itex]cosx[/itex] and [itex]secx[/itex] took one "x" from the original numerator?
 
  • #4
JasonHathaway said:
Did it multiply the numerator and denominator by [itex]\frac{cosx}{cosx}[/itex], which is [itex]cosx secx[/itex], and then both of [itex]cosx[/itex] and [itex]secx[/itex] took one "x" from the original numerator?
Yes.

In other words, ##\ \cos(x)\cdot\sec(x) = 1 \ .##
 

FAQ: Integration of x^2/(xsinx+cosx)^2

1. What is the purpose of integrating x^2/(xsinx+cosx)^2?

The purpose of integrating x^2/(xsinx+cosx)^2 is to find the area under the curve of the given function. Integration is a mathematical operation that is used to find the total accumulation of a quantity over a given interval. In this case, we are finding the total area under the curve of the function x^2/(xsinx+cosx)^2.

2. Is there a specific method or formula for integrating x^2/(xsinx+cosx)^2?

Yes, there are several methods and formulas for integrating x^2/(xsinx+cosx)^2. One common method is to use the substitution technique, where we substitute u = xsinx+cosx and then integrate using the u-substitution formula. Another method is to use integration by parts, where we split the function into two parts and use the integration by parts formula to solve for the integral.

3. What are the common mistakes to avoid when integrating x^2/(xsinx+cosx)^2?

One common mistake to avoid is forgetting to apply the chain rule when using the substitution method. Another mistake is to forget to include the constant of integration when solving the integral. It is also important to be careful with algebraic manipulations and simplifications, as they can lead to incorrect results.

4. Can the integration of x^2/(xsinx+cosx)^2 be solved analytically?

Yes, the integration of x^2/(xsinx+cosx)^2 can be solved analytically. Although it may require multiple steps and applications of different integration techniques, it is possible to find an exact analytical solution for the integral.

5. Are there any real-world applications of integrating x^2/(xsinx+cosx)^2?

Yes, there are many real-world applications of integrating x^2/(xsinx+cosx)^2. For example, this integral can be used to calculate the total energy of a vibrating string or to find the electric field around a charged ring. It can also be used in physics and engineering problems involving oscillations and vibrations.

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