(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

∫√(a^{2}- x^{2}) dx

2. Relevant equations

3. The attempt at a solution

What I've understood so far is its a substitution integral.

Through the instruction of my lecture notes, I have

∫√(a^{2}- x^{2}) dx Let u = sin^{-1}(x/a)

x = a*sin(u)

dx = a*cos(u)

This means that √(a^{2}- x^{2}) = √(a^{2}- a^{2}sin^{2}(x))

Then as the trig identity of √(cos^{2}(u) + sin^{2}(u)) = 1.

You can rearrange it to give cos(u) = √(1-sin^{2}(u))

So √(a^{2}- a^{2}sin^{2}(u)) = √(a^{2}* cos^{2}(u)) = a*cos(u)

Now there is my problem, A lot of other sources say that rather than ∫a*cos(u) du

it is ∫a^{2}* cos^{2}(u) du

and I don't understand why.

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# Integral of √(a^2 - x^2)

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