CBSE Class 12-science Answered

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

Asked by Adithyan Sp | 24 Feb, 2016, 06:10: PM

Expert Answer

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