I don't understand this integral

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    Integral
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Discussion Overview

The discussion revolves around understanding the integration of functions involving variables and constants, specifically focusing on the integral of ysin(xy)dx and the integration of x(y^2 - x^2)^(1/2). Participants seek clarification on the steps involved in these integrals and the reasoning behind certain constants in the results.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the integral ysin(xy)dx = -cos(xy) and seeks help in understanding how to eliminate y from the equation.
  • Another participant suggests finding the derivative of -cos(xy) to clarify the role of y in the integral.
  • A different participant assumes y is a constant and proposes a substitution z=xy to facilitate the integration process, while cautioning about changing limits of integration.
  • Several participants inquire about the integration of x(y^2 - x^2)^(1/2), with one participant stating their TA provided an answer of (-1/3)((y^2 - x^2)^(3/2)), but they do not understand the origin of the (-1/3) factor.
  • Another participant questions the lack of substitution rules in the integration process and asks for the work leading to the provided answer.
  • One participant shares their intermediate steps in the integration process but expresses uncertainty about the validity of a particular step involving (-1/x^2).
  • A later reply indicates confusion about the integration variable, noting it was not clearly stated that the integration was to be done with respect to x.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and confusion regarding the integration processes discussed. There is no consensus on the correct approach or resolution of the questions raised, indicating multiple competing views and unresolved issues.

Contextual Notes

Participants have not clearly defined assumptions regarding the variables involved, particularly the treatment of y as a constant. There are also unresolved steps in the integration processes that contribute to the confusion.

jkh4
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I don't understand for integral ysin(xy)dx = -cos(xy) for a=1 b=2. I know sin's integral is cos, but I don't understand how to eliminated the y in the left equation so it become the right equation. Please help!
 
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What is the derivative to -cos(xy)?

That should help to where the y is going.
 
jkh4 said:
I don't understand for integral ysin(xy)dx = -cos(xy) for a=1 b=2. I know sin's integral is cos, but I don't understand how to eliminated the y in the left equation so it become the right equation. Please help!

I am assuming that the a and b are the limits of integration.
And I am assuming that y is a constant here (it's independent of x). Then this is the simplest type of substitution: just define a new variable z=xy. What is dx then? You should then integrate easily (watch out about changing the limits of integration though if you leave your answer in terms of z).
 
what about this one?

how do you integrate x(y^2 - x^2)^(1/2)? my TA says the answer is (-1/3)((y^2 - x^2)^(3/2)) but i don't get where is the (-1/3) comes from...
 
jkh4 said:
what about this one?

how do you integrate x(y^2 - x^2)^(1/2)? my TA says the answer is (-1/3)((y^2 - x^2)^(3/2)) but i don't get where is the (-1/3) comes from...

Did you not do any substitution rules or anything?

Where is the work for this? Follow the work and it should be clear where it came from.
 
this is the process i got so far

(x^2/2)((y^2-x^2)^(3/2))/(3/2)(-1/x^2)

but one thing i don't understand, for the (-1/X^2), is this a proper intergral step?
 
jkh4 said:
this is the process i got so far

(x^2/2)((y^2-x^2)^(3/2))/(3/2)(-1/x^2)

but one thing i don't understand, for the (-1/X^2), is this a proper intergral step?

What?

Where does all this come from?
 
nevermind , i got it
 
Last edited:
One thing that was causing confusion throughout this thread- it was never stated that the integration was to be done with respect to x!
 

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