Applied Calculus Help

Hi,

Can anyone explain how to integrate (1-x^2)^1/2 using integration by parts as a solution I looked at is pretty confusing?

Thanks

Why by parts? The usual (famous) way is with a trig substitution. Or a sketch and a bit of geometry.

Idk my lecturer wants us to know this way too

Idk my lecturer wants us to know this way too

Just for completeness, a quick google this morning gave
https://math.stackexchange.com/questions/533082/integral-of-sqrt1-x2-using-integration-by-parts
and the multiplication by 1, then by parts, then trig substitution/spot integral was the way I was imagining last night. Whether you argue that spotting you can get a standard integral for arcsin is not doing trig substitution is debatable. You know the solution must contain an arcsin as the sector area is based on angle but the integral limits are based on side lengths/ratios so there must be an inverse trig kicking around in there somewhere.

A bit of head scratching this morning gave another by parts by doing dots on the original integrand then by parts on the two terms but you end up doing a more complex trig substitution at the end.

Thank you so much for this solution! I am still very confused tho haha but I will just stick to the trig substitution method. It won't let me upload the solution too:/

WHat are you confused about (in the stackexchange solution)?

The I bit