# Derivative and integral (confusing part)

by daivinhtran
Tags: confusing, derivative, integral
 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)
 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
 Homework Sci Advisor HW Helper Thanks P: 9,936 Consider the possibility that the two solutions are really the same, only differing by a constant.
P: 70
Derivative and integral (confusing part)

 Quote by haruspex 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???
 Homework Sci Advisor HW Helper Thanks P: 9,936 Yes. Can you see how? Hint: sin θ = cos(π/2-θ)
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P: 39,689
 Quote by daivinhtran Do you mean that both my and my teacher answers are right???
Wouldn't that be terrible!
P: 108
 Quote by daivinhtran 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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