Question

Integration Problem
Integral sqrt(1-x^2) dx from -1 to 1
Please give me the full solutions
Thank in advanced.

Answer 1

\[\int\limits \sqrt{1-x^2} \, dx?\]

Answer 2

Yes from -1 to 1

Answer 3

let x=sinu
dx=cosu du

Answer 4

\[\sqrt{1-x^2} =\sqrt{1-\sin ^2(u)}\text{ = }\cos (u)\text{ and }u=\sin ^{-1}(x)\]

Answer 5

\[\int\limits \cos ^2(u) \, du\]
Half angle formula
\[\int\limits \left(\frac{1}{2} \cos (2 u)+\frac{1}{2}\right) \, du\]
\[\text{}=\int\limits \frac{1}{2} \, du+\frac{1}{2}\int\limits \cos (2 u) \, du\]
\[\text{For} ~\cos (2 u)\text{, substitute }k=2 u\text{ and }dk=2d u:\]
\[\frac{1}{4}\int\limits \cos (k) \, dk+\int\limits \frac{1}{2} \, du\]
\[\frac{\sin (k)}{4}+\frac{u}{2}\]
\[\Huge [\frac{1}{2} \sqrt{1-x^2} x+\frac{1}{2} \sin ^{-1}(x)]_{-1}^{1}\]

Answer 6

Can I ask you some thing , integral sqrt(1-x^2) dx from -1 to 1 = pi/2

Answer 7

yes

Answer 8

integral sqrt(1-x^2) dx from -1 to 1 = pi/2

Answer 9

Oo thx

Answer 10

U = sin^-1 (x) ???