Trig Substitution for Integrating (y-x)/(a^2(y)-a^2(x))^b - Homework Help

[sara_87]

Homework Statement

I want to integrate:

$$\int^y_0\frac{y-x}{(a^2(y)-a^2(x))^b}dx$$

Homework Equations

a²(y) means that a is a function of y. similarly for [a(x)]². so [a(x)]² is a functions that depends on x.

The Attempt at a Solution

I tried integration by parts:
let
u=y-x so u'=-1
$$v'=(a^2(y)-a^2(x))^{-b}$$
and now i am struggling to integrate v'.

 

[g_edgar]

Is this the whole problem? Or are you, perhaps, supposed to do an additional computation with this integral? I ask because (since you don't know what function a is) you won't get any good expression for this integral by itself.

 

[Mark44]

I think I would start with a trig substitution first. Rather than remembering which formulas go with which situations, I draw a right triangle and label the sides and hypotenuse. I would label the hypotenuse a(y), and either of the other two sides as a(x). That leaves sqrt(a(y)^2 - a(x)^2) for the other side.

 

[sara_87]

Is this the whole problem? Or are you, perhaps, supposed to do an additional computation with this integral? I ask because (since you don't know what function a is) you won't get any good expression for this integral by itself.

this is the whole problem. the answer should be interms of the function a.

why would drawing a triangle help with choosing a substitution?

 

[Mark44]

Drawing a picture of a triangle is helpful when you're doing trig substitution, because it helps you establish the relationships between your substitution variable and the variables in your problem. You might be able to get by without drawing a picture, but you're probably more prone to making a mistake.

 

[sara_87]

i see what you mean.

so how about the substitution: a(x)=a(y)cos(x)
?

 

[Mark44]

Yes, that works. If you are using the triangle drawing, that would correspond to labelling the horizontal leg a(x) and the hypotenuse a(y).