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I'm working through Apostol's Calculus. I have attached the problem. I need to derive the formula integrating by parts.

It is not a hard problem, but I can't seem to understand how on earth the author came up with that expression.

I take f(x) = (a^2 - x^2)^n, so f'(x)=-2nx(a^2 - x^2)^(n-1).

Take g'(x)=1, so g(x)=x.

Plug the expressions in. We get another integral which could be easily solved with integration by substitution and I get the final expression of x(a^2 - x^2)^n - (a^2 - x^2)^n which does make sense if we check it by taking derivatives of both sides.

On the other hand, if I try to take the derivative of the given expression, I don't really get the initial result. So, am I missing something or what?

EDIT: OK, I've managed to see why his expression makes sense after all. but still, do not see how could I get there. Would my initial picks of f's and g' for integration by parts allow me to get there or should I pick something else? thanks!

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# Homework Help: Integration by substitution

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