laurellee27
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Show your working for the following integral by choosing a suitable substitution.
[tex]\int\limits{\sqrt{1-x^{2} } } \, dx[/tex]

Let x = sinu ∴ dx/du = cosu
[tex]\int\limits{\sqrt{1-x^{2} } cosu} \, du[/tex]
[tex]\int\limits{cosu^{2}} \, du[/tex]
I wanted to try integration by substitution, but I'm not sure if I'm going in the right direction.

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MathPhys

Answer:

½ asin x + ½ x √(1−x²) + C

Step-by-step explanation:

You're going in the right direction.  The next step is to use a power reduction formula:

∫ cos² u du

∫ (½ + ½ cos(2u)) du

½ ∫ du + ½ ∫ cos(2u) du

½ ∫ du + ¼ ∫ 2 cos(2u) du

½ u + ¼ sin(2u) + C

Next, we use double angle formula:

½ u + ½ sin u cos u + C

x = sin u, so u = asin x, and cos u = √(1−x²).

½ asin x + ½ x √(1−x²) + C

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