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Integral of x^3/square root of 1-x^2

Asked by Adithyan Sp 24th February 2016, 6:10 PM
Answered by Expert
Answer:

 

begin mathsize 16px style straight I space equals integral fraction numerator straight x cubed over denominator square root of 1 minus straight x squared end root end fraction dx  substitute space straight x squared space equals space straight u 2 xdx space equals space du  straight I space equals integral fraction numerator straight u over denominator 2 square root of 1 minus straight u end root end fraction du  substitute space square root of 1 minus straight u end root space equals space straight v straight u space equals space 1 minus straight v squared du space equals space left parenthesis negative 2 straight v right parenthesis dv  straight I space equals 1 half integral negative 2 left parenthesis 1 minus straight v squared right parenthesis dv equals 1 half open square brackets negative 2 open parentheses straight v minus straight v cubed over 3 close parentheses close square brackets plus straight C  resubstitute space square root of 1 minus straight u end root space equals space straight v  straight I space equals 1 half open square brackets negative 2 open parentheses square root of 1 minus straight u end root minus open parentheses square root of 1 minus straight u end root close parentheses cubed over 3 close parentheses close square brackets plus straight C  resubstitute space straight x squared space equals space straight u  straight I space equals 1 half open square brackets negative 2 open parentheses open parentheses square root of 1 minus straight x squared end root close parentheses minus open parentheses square root of 1 minus straight x squared end root close parentheses cubed over 3 close parentheses close square brackets plus straight C equals 1 third open parentheses 1 minus straight x squared close parentheses to the power of 3 over 2 end exponent minus square root of 1 minus straight x squared end root plus straight C end style

Answered by Expert 24th February 2016, 6:25 PM
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