- #1

- 124

- 0

## Homework Statement

integral of [ x*sqrt[1-x

^{2}] +

^{1}/

_{2 }- x

^{2}/

_{2}] dx - with the limits of 1 and 0

the last two bits are easy but i just wanted to confirm what i have done for the first bit is correct...ive tried using a method that takes less time but i don't know if its right...

## Homework Equations

sin(x)^2 + cos(x)^2 = 1

cos(arcsinx) = sqrt(1-x

^{2})

## The Attempt at a Solution

for the first bit:

x = sinu

dx = cosu du

1-x

^{2}= cos(u)^2

so it becomes: sinu(cos(u)

^{2})cosu du

sinu du = -dcosu

so :- -cos(u)

^{3}dcosu

integrate this beacomes: -cos(u)

^{4}/4

which then becomes

-cos(arcsinx)

^{4}/4 which is just: -(1-x

^{2})

^{2}/4

so putting this together with the 2nd and 3rd part gives

-(1-x

^{2})

^{2}/4 + x/2 - x

^{3}/6 with limts 1&0

which equals 1/3

so is this right?