Evaluate the following:
(i) `int sqrt(1+sin((1)/(2)x))dx`
(ii) `int(x^(2))/(sqrt(1-x))dx`

Answer

(i) `4 "sin"(x)/(4)-4"cos"(x)/(4)+c`
(ii) `-2{sqrt(1-x)-(2)/(3)(1-x)^(3//2)+(1)/(5)(1-x)^(5//2)}+c`

Explanation

(i) Let `I=int sqrt(1+"sin"(x)/(2))dx`
`=int sqrt("cos"^(2)(x)/(4)+"sin"^(2)(x)/(4)+2 "sin"(x)/(4)"cos"(x)/(4))dx`
`=int("cos"(x)/(4)+"sin"(x)/(4))dx=4"sin"(x)/(4)-4"cos"(x)/(4)+c`
(ii) Let `I=int(x^(2))/(sqrt(1-x))dx`
Put `1-x=t^(2) rArr -dx = 2"t dt"`
`therefore" "I=int((1-t^(2))^(2)*(-2t))/(t)dt`
`=-2 int(1-2t^(2)+t^(4))dt`
`=-2(t-(2t^(3))/(3)+(t^(4))/(5))+c`
`=-2{sqrt(1-x)-(2)/(3)(1-x)^(3//2)+(1)/(5)(1-x)^(5//2)}+c`