Understanding Properties of Integrals: How to Simplify an Integral

[Mrencko]

Homework Statement

i am studyng, for my multivariable calculus course, and i get this integral, the problem is, i don't know how the simplify the integral that way.

Homework Equations

i will put the integral in a very HD screen shoot, yes a litle one not the entire screen

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The Attempt at a Solution

i used the symetrical property of definite integrals, and get stuck when i check the simplification they did in the book, i need help to know how to continue, or what property they used. i got the second line OK, then to the thirth line i got stuck

 

[drvrm]

i used the symetrical property of definite integrals, and get stuck when i check the simplification they did in the book, i need help to know how to continue, or what property they used. i got the second line OK, then to the thirth line i got stuck

you can try substitution method.
hint convert it to sqrt(1+z^2) dz and then apply substitution.

 

[Mrencko]

hi tanks for the answer, but if i do the sustitution how the 1/d ends out of the integral?

 

[Mrencko]

sorry 1/p

 

[drvrm]

hi tanks for the answer, but if i do the sustitution how the 1/d ends out of the integral?

do not worry as you are integrating over x.

 

[Mrencko]

forgot it i have done z=x/2p
dz=1/2p
then... 2(1/2p)=1/p outside
now my doubt is how to make squart 4p2 + x2?

 

[Mrencko]

ok now i got 1/p∫√(1+z2)dz how i should proceed to make that into this 1/p∫√(4p2+x2)dx
its like somehow, they put the (x/2p)2=x2/4p2 into a sum, but only happens in logaritm properties

 

[drvrm]

ok now i got 1/p∫√(1+z2)dz how i should proceed to make that into this 1/p∫√(4p2+x2)dx

actually the idea is to get a standard form and then use the results from table of integrals

 

[SteamKing]

Homework Statement

i am studyng, for my multivariable calculus course, and i get this integral, the problem is, i don't know how the simplify the integral that way.

Homework Equations

i will put the integral in a very HD screen shoot, yes a litle one not the entire screenView attachment 98661 [/B]

The Attempt at a Solution

i used the symetrical property of definite integrals, and get stuck when i check the simplification they did in the book, i need help to know how to continue, or what property they used. i got the second line OK, then to the thirth line i got stuck

What happens when you put ##1 + (\frac{x}{2p})^2## over a common denominator? That's the step you're missing: how to add a fraction to a whole number.

 

[Mrencko]

Hi, so you mean 2p/2p +(x/2p)^2??

 

[Mrencko]

What happens when you put ##1 + (\frac{x}{2p})^2## over a common denominator? That's the step you're missing: how to add a fraction to a whole number.

thanks for reply

 

[SteamKing]

Hi, so you mean 2p/2p +(x/2p)^2??

Close, but you want to get the quantity ##(\frac{x}{2p})^2## added to 1.

 

[Mrencko]

Well, following your previous hint, a way to add a fraction number, to a whole y should do the following: 1 + x^2/4p^2=(4p^2+x^2)/4p^2
It's ok?

 

[SteamKing]

Well, following your previous hint, a way to add a fraction number, to a whole y should do the following: 1 + x^2/4p^2=(4p^2+x^2)/4p^2
It's ok?

That looks OK. You should be able to simplify the original integral as shown.

 

[Mrencko]

Oh my god it was so obvious, many thanks now looks like squareroot((1/4p^2)(4p^2 +x^)