Integral of sqrt(9-x^2) (substitution)

Опубликовано: 13 Март 2026
на канале: Integrals ForYou

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Integral of sqrt(9-x^2) - How to integrate it step by step using the trig substitution method!

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Derivative of (9/2)arcsin(x/3) + (x/2)sqrt(9-x^2) =    • Derivative of (9/2)arcsin(x/3) + (x/2)sqrt...

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🚶 𝐒𝐭𝐞𝐩𝐬
00:00 Take out 9 as common factor
00:15 Rewrite expression
00:31 Substitution: x/3=sin(u)
00:41 Differentiate in both sides
00:57 Substitute 1-(x/3)^2 and dx
01:33 Rewrite expression
01:44 Use the cos^2(u)=(1/2)(1+cos(2u)) formula
02:02 Split into two integrals
02:20 Integrate u and cos(2u)
02:40 Use the sin(2u)=2sin(u)cos(u) formula
02:58 Undo substitution: u in terms of x
03:29 Use the sin(arcsin(x/3)) and cos(arcsin(x/3)) formulas
03:52 Rewrite expression
04:51 Add integration constant +C
04:59 Final answer!

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