I have been trying to figure out this problem for hours. I'm probably doing something ridiculously stupid, but, alas, I truly have no clue where to go next.
PROBLEM: ?(4-x[sup:32hjfs8g]2[/sup:32hjfs8g])[sup:32hjfs8g]1/2[/sup:32hjfs8g]
One of the many ways I've attempted (this way got me closest):
?(4(1-(x[sup:32hjfs8g]2[/sup:32hjfs8g]/4)))[sup:32hjfs8g]1/2[/sup:32hjfs8g] - Take out a 4
-------------------------------------------------------------------
//BEGIN OFF TO THE SIDE WORK
(x[sup:32hjfs8g]2[/sup:32hjfs8g]/4) = sin[sup:32hjfs8g]2[/sup:32hjfs8g]?
SQUARE BOTH SIDES
(x/2) = sin? --> THEREFORE --> ? = arcsin(x/2)
x = 2sin?
dx = 2cos? d?
//END OFF TO THE SIDE WORK
-------------------------------------------------------------------
SUBSTITUTE IN NEW STUFF
?((4(1-sin[sup:32hjfs8g]2[/sup:32hjfs8g]?))[sup:32hjfs8g]1/2[/sup:32hjfs8g])(2cos?)(d?)
USE TRIG IDENTITY
?((4(cos[sup:32hjfs8g]2[/sup:32hjfs8g]?))[sup:32hjfs8g]1/2[/sup:32hjfs8g])(2cos?)(d?)
EXECUTE SQUARE ROOT FUNCTION
?(2(cos?))(2cos?)(d?)
?(4(cos[sup:32hjfs8g]2[/sup:32hjfs8g]?)(d?)
TAKE OUT 4
4?(cos[sup:32hjfs8g]2[/sup:32hjfs8g]?)(d?)
USE TRIG IDENTITY
4?(1/2)(1+cos2?)(d?)
TAKE OUT (1/2)
2?(1+cos2?)(d?)
DISTRIBUTE
2?(d?)+(cos2?)(d?)
SEPARATE INTO TWO PARTS
2?(d?) + 2?(cos2?)(d?)
INTEGRATE
2? + sin2?
RESUB in (?=arsin(x/2))?
sin2? + 2arcsin(x/2)
And I have no clue how to get from there, to the answer given here:
2arcsin(x/2) + ((1/2)(x))(((4-x[sup:32hjfs8g]2[/sup:32hjfs8g]))[sup:32hjfs8g]1/2[/sup:32hjfs8g]) + C
PROBLEM: ?(4-x[sup:32hjfs8g]2[/sup:32hjfs8g])[sup:32hjfs8g]1/2[/sup:32hjfs8g]
One of the many ways I've attempted (this way got me closest):
?(4(1-(x[sup:32hjfs8g]2[/sup:32hjfs8g]/4)))[sup:32hjfs8g]1/2[/sup:32hjfs8g] - Take out a 4
-------------------------------------------------------------------
//BEGIN OFF TO THE SIDE WORK
(x[sup:32hjfs8g]2[/sup:32hjfs8g]/4) = sin[sup:32hjfs8g]2[/sup:32hjfs8g]?
SQUARE BOTH SIDES
(x/2) = sin? --> THEREFORE --> ? = arcsin(x/2)
x = 2sin?
dx = 2cos? d?
//END OFF TO THE SIDE WORK
-------------------------------------------------------------------
SUBSTITUTE IN NEW STUFF
?((4(1-sin[sup:32hjfs8g]2[/sup:32hjfs8g]?))[sup:32hjfs8g]1/2[/sup:32hjfs8g])(2cos?)(d?)
USE TRIG IDENTITY
?((4(cos[sup:32hjfs8g]2[/sup:32hjfs8g]?))[sup:32hjfs8g]1/2[/sup:32hjfs8g])(2cos?)(d?)
EXECUTE SQUARE ROOT FUNCTION
?(2(cos?))(2cos?)(d?)
?(4(cos[sup:32hjfs8g]2[/sup:32hjfs8g]?)(d?)
TAKE OUT 4
4?(cos[sup:32hjfs8g]2[/sup:32hjfs8g]?)(d?)
USE TRIG IDENTITY
4?(1/2)(1+cos2?)(d?)
TAKE OUT (1/2)
2?(1+cos2?)(d?)
DISTRIBUTE
2?(d?)+(cos2?)(d?)
SEPARATE INTO TWO PARTS
2?(d?) + 2?(cos2?)(d?)
INTEGRATE
2? + sin2?
RESUB in (?=arsin(x/2))?
sin2? + 2arcsin(x/2)
And I have no clue how to get from there, to the answer given here:
2arcsin(x/2) + ((1/2)(x))(((4-x[sup:32hjfs8g]2[/sup:32hjfs8g]))[sup:32hjfs8g]1/2[/sup:32hjfs8g]) + C