Integration problem (algebraic+trigonometric function)

In summary, the conversation discusses finding the integral of the function (x^2 + cos^2)(cosec^2) / (1+x^2). The individual attempted to solve the problem using various methods, but struggled to simplify the integrand. Eventually, it was suggested to use a trig identity, which led to a simple solution.
  • #1

Krushnaraj Pandya

Gold Member
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Homework Statement


find integral of the function- (x^2 + cos^2)(cosec^2) / (1+x^2)

2. The attempt at a solution
I noticed the denominator is the derivative of Arctan(x), I tried integrating by parts with various choices for 1st and second function but all of them end up being more complicated, I tried factoring in sec^2(x) which also didn't go a long way. I would appreciate some help to proceed further with this integral
 
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  • #2
Krushnaraj Pandya said:
I tried factoring in sec^2(x) which also didn't go a long way.
What does this mean?
 
  • #3
vela said:
What does this mean?
It means I tried multiplying and dividing by sec^2(x), but still couldn't integrate it
 
  • #4
You're overthinking it. Multiply the numerator out and go from there.
 
  • #5
vela said:
You're overthinking it. Multiply the numerator out and go from there.
you mean separating numerator and denominator as two different functions and then integrate by parts? I already tried that
 
  • #6
No, you just want to simplify the integrand first.
 
  • #7
vela said:
No, you just want to simplify the integrand first.
what did you mean by "multiply the numerator out?" multiply cosec^2 inside?
 
  • #8
Distribute the factor of ##csc^2 x## into the sum.
 
  • #9
vela said:
Distribute the factor of ##csc^2 x##.
done already. ## cosec^2(x) x^2 + cot^2(x) ## in the numerator
then i separated both terms but i'd have to integrate both of them by parts which is really long
 
  • #10
You haven't really simplified the integrand. To me, the obvious thing to try here is a trig identity. You have csc^2 and cot^2...a natural choice comes to mind. See what happens.
 
  • #11
vela said:
You haven't really simplified the integrand. To me, the obvious thing to try here is a trig identity. You have csc^2 and cot^2...a natural choice comes to mind. See what happens.
It was so easy! I really did overthink it. Thanks a lot though :)
 
1.

What is an integration problem involving algebraic and trigonometric functions?

An integration problem involving algebraic and trigonometric functions is a mathematical problem where the goal is to find the antiderivative of a combination of functions involving both algebraic and trigonometric functions. This can be done using techniques such as substitution, integration by parts, or trigonometric identities.

2.

What are some common techniques for solving integration problems involving algebraic and trigonometric functions?

Some common techniques for solving integration problems involving algebraic and trigonometric functions include substitution, integration by parts, and using trigonometric identities. It is also important to have a good understanding of the properties of algebraic and trigonometric functions and how they behave under integration.

3.

What are the steps for solving an integration problem involving algebraic and trigonometric functions?

The general steps for solving an integration problem involving algebraic and trigonometric functions are: 1) Identify the type of function and choose an appropriate integration technique, 2) Apply the chosen technique to find the antiderivative, 3) Check the solution by taking the derivative to see if it matches the original function, and 4) Add the constant of integration to the final answer.

4.

What are some common mistakes to avoid when solving integration problems involving algebraic and trigonometric functions?

Some common mistakes to avoid when solving integration problems involving algebraic and trigonometric functions include: forgetting to add the constant of integration, making mistakes with algebraic manipulations, and forgetting to use trigonometric identities when necessary. It is also important to carefully check the solution by taking the derivative to avoid any errors.

5.

How can I practice and improve my skills in solving integration problems involving algebraic and trigonometric functions?

The best way to practice and improve your skills in solving integration problems involving algebraic and trigonometric functions is to solve a variety of problems using different techniques. You can find practice problems online, in textbooks, or create your own. It is also helpful to review the properties of algebraic and trigonometric functions and their integration rules. Additionally, seeking help from a tutor or teacher can also be beneficial in improving your skills.

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