So I am attempting the question and I\'m stuck part way through. I have evaluate

Question

So I am attempting the question and I'm stuck part way through. I have evaluated the indefinite integral of [sin^(-1)x dx] and retrieved the answer [x sin^(-1) x + sqrt(1+x^2) + C] using integration by parts and substitution. However, I can't seem to get a clean answer from evaluating the definite integral. Also, because I used u-substitution, I know I am to substitute the x values 0 and (1/2) in order to obtain the corresponding u values (3/4 and 1), but since I only used u-substitution to find half of the equation, namely the square root of 1-x^2, do I only solve using the values (3/4) and 1 for this part of the equation?

Explanation / Answer

u = arcsin(x)
sin(u) = x
cos(u) * du = dx
du = dx / cos(u)
du = dx / sqrt(1 - sin(u)^2)
du = dx / sqrt(1 - x^2)

dv = dx
v = x

int(u * dv) = u * v - int(v * du)
x * arcsin(x) - int(x * dx / sqrt(1 - x^2))

u = 1 - x^2
du = -2x * dx

x * arcsin(x) - int((-1/2) * (-2) * x * dx / sqrt(1 - x^2)
{I hope you can see why I put (-1/2) * (-2) in there, because it multiplies out to 1}
x * arcsin(x) - (-1/2) * int((-2x * dx) / sqrt(1 - x^2))
x * arcsin(x) + (1/2) * int((-2x * dx) / sqrt(1 - x^2))
x * arcsin(x) + (1/2) * int(du / sqrt(u))
x * arcsin(x) + (1/2) * int(u^(-1/2) * du)
x * arcsin(x) + (1/2) * (1/(-1/2 + 1)) * u^(-1/2 + 1) + C =>
x * arcsin(x) + (1/2) * 2 * u^(1/2) + C =>
x * arcsin(x) + sqrt(u) + C =>
x * arcsin(x) + sqrt(1 - x^2) + C

x = 0 , 1/2

(1/2) * arcsin(1/2) + sqrt(1 - (1/2)^2) - 0 * arcsin(0) - sqrt(1 - 0^2) =>
(1/2) * (pi/6) + sqrt(1 - 1/4) - sqrt(1) =>
pi/12 + sqrt(3/4) - 1 =>
pi/12 + sqrt(3)/2 - 1 =>
pi/12 + 6sqrt(3)/12 - 12/12 =>
(pi + 6 * sqrt(3) - 12) / 12

That's as nice as it's getting.

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