integral clarification

[mcwang719]

hello, i've been looking at this example for a while and can't seem to understand what's going on. it seems like this is a u-substitution problem. where does x= sin come from. are they even using u-sub? can someone please help me understand what's going on?thanks!!!



∫2√(1-x^2)dx, let x=sin θ, so dx=cosθdθ
= ∫2cos^(2)θdθ = ∫(1+cos2θ)dθ
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[galactus]

It's trigononmetric substitution.


\(\displaystyle \L\\2\int\sqrt(1-x^{2})dx\)

Letting \(\displaystyle x=sin({\theta})\) and \(\displaystyle dx=cos({\theta})d{\theta}\)

\(\displaystyle \L\\2\int\sqrt{1-sin^{2}({\theta})}cos({\theta})d({\theta})\)

Since \(\displaystyle \L\\1-sin^{2}({\theta})=cos^{2}({\theta})\)

We have:

\(\displaystyle \L\\2\int\sqrt{cos^{2}({\theta})}cos({\theta})d{\theta}\)

\(\displaystyle \L\\2\int{cos^{2}({\theta})}d{\theta}\)

Since, \(\displaystyle \L\\cos^{2}({\theta})=\frac{1}{2}(1+cos(2{\theta})\)

Then, \(\displaystyle \L\\2cos^{2}({\theta})=1+cos(2{\theta})\)

Your integral:

\(\displaystyle \L\\\int(1+cos(2{\theta}))d{\theta}\)

Now, finish the integration. The worst part is over.