Integral of x(sqrt(1-x^2)) + 1/2 -x^2/2 IS IT RIGHT?

[indie452]

Homework Statement

integral of [ x*sqrt[1-x²] + ¹/₂
- x²/₂] 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-x²)

The Attempt at a Solution

for the first bit:

x = sinu
dx = cosu du
1-x² = 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-x²)²/4

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

-(1-x²)²/4 + x/2 - x³/6 with limts 1&0

which equals 1/3

so is this right?

 

[Cyosis]

Use the right tool for the right situation. Trigonometric substitution certainly works, but it takes you on quite a detour. To check if your answer is correct you simply differentiate it so let's do that.

$$\frac{d}{dx} \frac{-(1-x^2)^2}{4}=x(1-x^2) \neq x \sqrt{1-x^2}$$

You make an error very early on after you state that 1-x^2=cos(u)^2 you conclude in the next line that $\sqrt{1-x^2}=\cos^2u$. In other words you forgot to take the square root. The rest seems to be correct.

There is however a method that is way easier take u=1-x^2 and you're done.

 

[indie452]

as soon as you pointed out that error early on i felt like kicking myself yeah i see what to do now

(do you still get 1/3 as the answer cause I've just redone it and I am still getting the same answer with the limits in)

thanks

 

[Cyosis]

For the first term I get 1/3 for the second 1/2 and for the third -1/6. So 1/3+1/2-1/6=2/3

 

[tiny-tim]

Welcome to PF!

Hi indie452! Welcome to PF!

(have a square-root: √ )

There is however a method that is way easier take u=1-x^2 and you're done.

indie452, have you tried Cyosis's sub, u = √(1 - x²)?

It's much quicker, and it's a standard technique that you do need to learn.

 

[indie452]

yeah i tried it ... the reason i got the wrong answer the 2nd time was cause i messed up on subbing the limits but i got the right answer now