# Two easy integrals

• Dec 8th 2007, 08:29 AM
liyi
Two easy integrals
Integrate

$\displaystyle f(x)=\sqrt{1-x^2}$ & $\displaystyle h(x)=\sqrt{x^2-1}$
• Dec 8th 2007, 09:10 AM
curvature
Quote:

Originally Posted by liyi
Integrate

$\displaystyle f(x)=\sqrt{1-x^2}$ & $\displaystyle h(x)=\sqrt{x^2-1}$

Do the substitutions: x=sint for the first one and x=sect for the second.
• Dec 8th 2007, 09:16 AM
markcantdomaths
I'm not sure what the user above means but I'd;

Rearrange
$\displaystyle f'(x)=\sqrt{1-x^2}$
$\displaystyle f'(x)=\sqrt{1} - \sqrt{x^2}$
$\displaystyle f'(x)= 1^{1/2} - x$
Intergrate; add one to power, divide by new power and add c:
$\displaystyle f(x)= 2/3^{3/2} - 1 + c$
I don't think that can be simplified
• Dec 8th 2007, 09:23 AM
topsquark
Quote:

Originally Posted by markcantdomaths
I'm not sure what the user above means but I'd;

Rearrange
$\displaystyle f'(x)=\sqrt{1-x^2}$
$\displaystyle f'(x)=\sqrt{1} - \sqrt{x^2}$

In general $\displaystyle \sqrt{a^2 + b^2} \neq a + b$

curvature's substitutions should work nicely.

-Dan
• Dec 8th 2007, 09:23 AM
curvature
Quote:

Originally Posted by markcantdomaths
Rearrange
$\displaystyle f'(x)=\sqrt{1-x^2}$
$\displaystyle f'(x)=\sqrt{1} - \sqrt{x^2}$

No the second equality is wrong.
• Dec 8th 2007, 09:55 AM
Krizalid
Quote:

Originally Posted by liyi
Integrate

$\displaystyle f(x)=\sqrt{1-x^2}$

Define the function $\displaystyle u(x) = x\sqrt {1 - x^2 } \implies u'(x) = \sqrt {1 - x^2 } - \frac{{x^2 }} {{\sqrt {1 - x^2 } }} = 2\sqrt {1 - x^2 } - \frac{1} {{\sqrt {1 - x^2 } }}.$

Make-up: $\displaystyle \sqrt {1 - x^2 } = \frac{1} {2}\left[ {u'(x) + \frac{1} {{\sqrt {1 - x^2 } }}} \right].$

Integrate $\displaystyle \int {\sqrt {1 - x^2 } \,dx} = \frac{1} {2}\left( {x\sqrt {1 - x^2 } + \arcsin x} \right) + k.$

Quote:

Originally Posted by liyi
Integrate

$\displaystyle h(x)=\sqrt{x^2-1}$

Consider the function $\displaystyle \varphi (x) = x\sqrt {x^2 - 1} \implies \varphi '(x) = \sqrt {x^2 - 1} + \frac{{x^2 }} {{\sqrt {x^2 - 1} }} = 2\sqrt {x^2 - 1} + \frac{1} {{\sqrt {x^2 - 1} }}.$

Make-up & integrate

$\displaystyle \int {\sqrt {x^2 - 1} \,dx} = \frac{1} {2}\left( {x\sqrt {x^2 - 1} - \int {\frac{1} {{\sqrt {x^2 - 1} }}\,dx} } \right)+k_1.$

For the remaining integral, substitute $\displaystyle u=x+\sqrt{x^2-1},$

$\displaystyle \int {\frac{1} {{\sqrt {x^2 - 1} }}\,dx} = \int {\frac{1} {u}\,du} = \ln \left| {x + \sqrt {x^2 - 1} } \right| + k_2.$

And we happily get $\displaystyle \int {\sqrt {x^2 - 1} \,dx} = \frac{1} {2}\left( {x\sqrt {x^2 - 1} - \ln \left| {x + \sqrt {x^2 - 1} } \right|} \right) + k.$
• Dec 8th 2007, 10:34 PM
angel.white
Quote:

Originally Posted by Krizalid
Define the function $\displaystyle u(x) = x\sqrt {1 - x^2 } \implies u'(x) = \sqrt {1 - x^2 } - \frac{{x^2 }} {{\sqrt {1 - x^2 } }} = 2\sqrt {1 - x^2 } - \frac{1} {{\sqrt {1 - x^2 } }}.$

Make-up: $\displaystyle \sqrt {1 - x^2 } = \frac{1} {2}\left[ {u'(x) + \frac{1} {{\sqrt {1 - x^2 } }}} \right].$

Integrate $\displaystyle \int {\sqrt {1 - x^2 } \,dx} = \frac{1} {2}\left( {x\sqrt {1 - x^2 } + \arcsin x} \right) + k.$

Consider the function $\displaystyle \varphi (x) = x\sqrt {x^2 - 1} \implies \varphi '(x) = \sqrt {x^2 - 1} + \frac{{x^2 }} {{\sqrt {x^2 - 1} }} = 2\sqrt {x^2 - 1} + \frac{1} {{\sqrt {x^2 - 1} }}.$

Make-up & integrate

$\displaystyle \int {\sqrt {x^2 - 1} \,dx} = \frac{1} {2}\left( {x\sqrt {x^2 - 1} - \int {\frac{1} {{\sqrt {x^2 - 1} }}\,dx} } \right)+k_1.$

For the remaining integral, substitute $\displaystyle u=x+\sqrt{x^2-1},$

$\displaystyle \int {\frac{1} {{\sqrt {x^2 - 1} }}\,dx} = \int {\frac{1} {u}\,du} = \ln \left| {x + \sqrt {x^2 - 1} } \right| + k_2.$

And we happily get $\displaystyle \int {\sqrt {x^2 - 1} \,dx} = \frac{1} {2}\left( {x\sqrt {x^2 - 1} - \ln \left| {x + \sqrt {x^2 - 1} } \right|} \right) + k.$

I understood everything except this step here:
$\displaystyle \sqrt {1 - x^2 } - \frac{{x^2 }} {{\sqrt {1 - x^2 } }} = 2\sqrt {1 - x^2 } - \frac{1} {{\sqrt {1 - x^2 } }}.$

• Dec 8th 2007, 10:49 PM
Jhevon
Quote:

Originally Posted by angel.white
I understood everything except this step here:
$\displaystyle \sqrt {1 - x^2 } - \frac{{x^2 }} {{\sqrt {1 - x^2 } }} = 2\sqrt {1 - x^2 } - \frac{1} {{\sqrt {1 - x^2 } }}.$

as in explain why it is true?

$\displaystyle \sqrt{1 - x^2} - \frac {x^2}{\sqrt {1 - x^2}} = \sqrt{1 - x^2} - \frac {x^2 + 1 - 1}{\sqrt {1 - x^2}}$

$\displaystyle = \sqrt{1 - x^2} + \frac {1 - x^2}{\sqrt {1 - x^2}} - \frac 1{\sqrt {1 - x^2}}$

$\displaystyle = \sqrt{1 - x^2} + \sqrt{1 - x^2} - \frac 1{\sqrt {1 - x^2}}$

$\displaystyle = 2 \sqrt{1 - x^2} - \frac 1{\sqrt {1 - x^2}}$
• Dec 8th 2007, 10:53 PM
angel.white
I see now, that's a smart way of doing business.