I don't understand this integral

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SUMMARY

The integral of ysin(xy)dx from a=1 to b=2 simplifies to -cos(xy) when y is treated as a constant independent of x. The substitution z=xy is crucial for simplifying the integral, allowing for straightforward integration. Additionally, the confusion regarding the integral x(y^2 - x^2)^(1/2) arises from the application of substitution rules, leading to the result of (-1/3)((y^2 - x^2)^(3/2)). Understanding the limits of integration and the variable of integration is essential for clarity in these problems.

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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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