Question

substitution for √(1-x^2) is ----- *

Answer 1

Step-by-step explanation:

As the integrand function is defined for

x

∈

[

−

1

,

1

]

, you can substitute:

x

=

sin

t

with

t

∈

[

−

π

2

,

π

2

]

d

x

=

cos

t

d

t

so the integral becomes:

∫

√

1

−

x

2

d

x

=

∫

√

1

−

sin

2

t

cos

t

d

t

=

∫

√

cos

2

t

cos

t

d

t

In the given interval

cos

t

is positive, so

√

cos

2

t

=

cos

t

:

∫

√

1

−

x

2

d

x

=

∫

cos

2

t

d

t

Now we can use the identity:

cos

2

t

=

1

+

cos

(

2

t

)

2

∫

√

1

−

x

2

d

x

=

∫

1

+

cos

(

2

t

)

2

d

t

=

∫

d

t

2

+

1

4

∫

cos

(

2

t

)

d

(

2

t

)

=

1

2

t

+

1

4

sin

2

t

=

1

2

(

t

+

sin

t

cos

t

)

To substitute back

x

we note that:

x

=

sin

t

for

t

∈

[

−

π

2

,

π

2

]

⇒

t

=

arcsin

x

cos

t

=

√

1

−

sin

2

t

=

√

1

−

x

2

Finally:

∫

√

1

−

x

2

d

x

=

1

2

(

arcsin

x

+

x

√

1

−

x

2

)

+

C