the derivative: 1/2 *( [sqrt(1-x^2)+x] * e^(arcsinx) )' = 1/2 *[ 1/2 *(-2x) /sqrt(1 - x^2)+1 + x/sqrt(1 - x^2) +1 ] * e^(arcsinx) = 1/2 *2e^(arcsinx) = e^(arcsinx)

OMG!!!!!!!!!!

/ e^(arcsinx)dx = x * e^(arcsinx) - / x * e^(arcsinx)/sqrt(1-x^2) dx = x * e^(arcsinx) + / 1/2 * e^(arcsinx) * 2 d sqrt(1 - x^2) = x * e^(arcsinx) + e^(arcsinx) * sqrt(1 - x^2) - / sqrt(1 - x^2)/ sqrt(1 - x^2) e^(arcsinx) dx = x * e^(arcsinx) + e^(arcsinx) * sqrt(1 - x^2) - / e^(arcsinx)dx then can get: / 2e^(arcsinx)dx = x * e^(arcsinx) + e^(arcsinx) * sqrt(1 - x^2) + C' then: / e^(arcsinx)dx = 1/2 * e^(arcsinx) [x + sqrt(1 - x^2)] + C or : let arcsinx = u, (-pi/2 < u < pi/2 ), then x = sinu, dx = cosu du , / e^(arcsinx) dx = / e^u * cos u * du = / cos u * d e^u = cos u * e^u + / e^u sin u du = cos u * e^u + / sin u d e^u = cos u * e^u + sin u * e^u - / e^u cos u du then can get : 2 / e^u cos u du = cos u * e^u + sin u * e^u + C' then : / e^u cos u du =1/2 [ cos u + sin u * ] * e^u + C = 1/2 [ sqrt(1 - x^2) + x] * e^(arcsinx) + C