Derivative and integral (confusing part)


by daivinhtran
Tags: confusing, derivative, integral
daivinhtran
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#1
Dec9-12, 11:12 PM
P: 70
We know that d(cos^-1 (x/a))/dx = -1/sqrt(a^2 - x^2) (assuming a and x are positive)

So...Why the integral of -1/sqrt(a^2 - x^2) is not equal to (cos^-1 (x/a)) + C??????????

Instead, my teacher says it has to be -(sin^-1 (x/a)) + C because integral of 1/sqrt(a^2 - x^2) is sin^-1 (x/a) + C.....(only put the negative sign in)

She doesn't really explain though...(when I ask)
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daivinhtran
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#2
Dec9-12, 11:24 PM
P: 70
However, when I use wolmframalpha, it gives a different solution rather than the sin^-1

http://www.wolframalpha.com/input/?i=+integrate+of+-1%2Fsqrt%28a^2+-+x^2%29+

Here the link
haruspex
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#3
Dec9-12, 11:49 PM
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Consider the possibility that the two solutions are really the same, only differing by a constant.

daivinhtran
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#4
Dec10-12, 07:31 PM
P: 70

Derivative and integral (confusing part)


Quote Quote by haruspex View Post
Consider the possibility that the two solutions are really the same, only differing by a constant.


Do you mean that both my and my teacher answers are right???
haruspex
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Dec10-12, 09:23 PM
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Yes. Can you see how? Hint: sin θ = cos(π/2-θ)
HallsofIvy
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Dec11-12, 08:08 PM
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Quote Quote by daivinhtran View Post
Do you mean that both my and my teacher answers are right???
Wouldn't that be terrible!
sahil_time
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#7
Dec17-12, 01:16 PM
P: 108
Quote Quote by daivinhtran View Post
We know that d(cos^-1 (x/a))/dx = -1/sqrt(a^2 - x^2) (assuming a and x are positive)

So...Why the integral of -1/sqrt(a^2 - x^2) is not equal to (cos^-1 (x/a)) + C??????????

Instead, my teacher says it has to be -(sin^-1 (x/a)) + C because integral of 1/sqrt(a^2 - x^2) is sin^-1 (x/a) + C.....(only put the negative sign in)

She doesn't really explain though...(when I ask)

Its because cos^-1(x) and sin^-1(x) are related by a constant.

cos^-1(x) + sin^-1(x) = ∏/2

So if you write
∫-1/√(a^2 - x^2) = cos^-1 (x/a) + C

I can use the relation and write

∫-1/√(a^2 - x^2) = ∏/2 - sin^-1(x/a) + C

Which is equal to

∫-1/√(a^2 - x^2) = - sin^-1(x/a) + C1

Where C1 is some arbitrary constant = C + ∏/2.

So its like haruspex said, solutions differ only by a constant.

Hope that Helps :)


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