# integral problem...

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• Apr 23rd 2009, 08:56 AM
Mathventure
integral problem...
Q: let f(x)=integral of e^(-t^2) limit from sin(x) to cos(x).Then find f'(pi/4)=
• Apr 23rd 2009, 10:52 AM
Jester
Quote:

Originally Posted by Mathventure
Q: let f(x)=integral of e^(-t^2) limit from sin(x) to cos(x).Then find f'(pi/4)=

$f(x) = \int_{\sin x}^{\cos x} e^{-t^2}\,dt = \int_a^{\cos x} e^{-t^2}\,dt + \int_{\sin x}^a e^{-t^2}\,dt = \int_a^{\cos x} e^{-t^2}\,dt - \int_a^{\sin x} e^{-t^2}\,dt$ so

$f'(x) = - e^{-\cos^2 x}\sin x - e^{-\sin^2 x}\cos x$

• Apr 23rd 2009, 06:51 PM
Mathventure
Quote:

Originally Posted by danny arrigo
$f(x) = \int_{\sin x}^{\cos x} e^{-t^2}\,dt = \int_a^{\cos x} e^{-t^2}\,dt + \int_{\sin x}^a e^{-t^2}\,dt = \int_a^{\cos x} e^{-t^2}\,dt - \int_a^{\sin x} e^{-t^2}\,dt$ so

$f'(x) = e^{-\cos^2 x}\sin x - e^{-\sin^2 x}\cos x$

after putting value i got zero but options available are
a)sqrt(1/e)
b)-sqrt(2/e)
c)sqrt(2/e)
d)-sqrt(1/e)
• Apr 23rd 2009, 07:02 PM
NonCommAlg
Quote:

Originally Posted by Mathventure
after putting value i got zero but options available are
a)sqrt(1/e)
b)-sqrt(2/e)
c)sqrt(2/e)
d)-sqrt(1/e)

that was just a typo and you should have noticed it. what danny meant was this: $f'(x) = -e^{-\cos^2 x}\sin x - e^{-\sin^2 x}\cos x.$
• Apr 23rd 2009, 07:04 PM
Mathventure
Quote:

Originally Posted by NonCommAlg
that was just a typo and you should have noticed it. what danny meant was this: $f'(x) = -e^{-\cos^2 x}\sin x - e^{-\sin^2 x}\cos x.$

but how he solved integral??
• Apr 23rd 2009, 07:09 PM
skeeter
you don't need to solve the integral ... use the 2nd FTC.

$f(x) = \int_{\sin{x}}^{\cos{x}} e^{-t^2} \, dt$

$f'(x) = -e^{-\cos^2{x}} \sin{x} - e^{-\sin^2{x}} \cos{x}$

$f'\left(\frac{\pi}{4}\right) = -e^{-\frac{1}{2}} \cdot \frac{\sqrt{2}}{2} - e^{-\frac{1}{2}} \cdot \frac{\sqrt{2}}{2}
$

$f'\left(\frac{\pi}{4}\right) = -\sqrt{\frac{2}{e}}
$
• Apr 23rd 2009, 07:12 PM
Mathventure
Quote:

Originally Posted by skeeter
you don't need to solve the integral ... use the 2nd FTC.

$f(x) = \int_{\sin{x}}^{\cos{x}} e^{-t^2} \, dt$

$f'(x) = -e^{-\cos^2{x}} \sin{x} - e^{-\sin^2{x}} \cos{x}$

$f'\left(\frac{\pi}{4}\right) = -e^{-\frac{1}{2}} \cdot \frac{\sqrt{2}}{2} - e^{-\frac{1}{2}} \cdot \frac{\sqrt{2}}{2}
$

$f'\left(\frac{\pi}{4}\right) = -\sqrt{\frac{2}{e}}
$

i was really silly that i forgot FTC....
• Apr 23rd 2009, 07:13 PM
NonCommAlg
Quote:

Originally Posted by Mathventure

but how he solved integral??

let $\int_a^x f(t) \ dt = g(x),$ where $a$ is a constant. by the fundamental theorem of calculus we have $g'(x) = f(x).$ now if $u(x),v(x)$ are differentiable functions, then:

$h(x)=\int_{u(x)}^{v(x)} f(t) \ dt = g(v(x)) - g(u(x)).$ so by the chain rule: $h'(x)=v'(x)g'(v(x)) - u'(x)g'(u(x))=v'(x)f(v(x))-u'(x)f(u(x)).$
• Apr 23rd 2009, 07:18 PM
derfleurer
Edit: It really takes too long typing latex sometimes. =p

Quote:

Originally Posted by Mathventure
but how he solved integral??

What's to solve?

$f(x) = \int_{sinx}^{cosx} e^{-t^2}dt$

If that's f(x), that little function in the integral looks like f'(x)

$e^{-(cosx)^2}(-sinx) - e^{-(sinx)^2}(cosx)$
• Apr 23rd 2009, 07:24 PM
Mathventure
Quote:

Originally Posted by NonCommAlg
let $\int_a^x f(t) \ dt = g(x),$ where $a$ is a constant. by the fundamental theorem of calculus we have $g'(x) = f(x).$ now if $u(x),v(x)$ are differentiable functions, then:

$h(x)=\int_{u(x)}^{v(x)} f(t) \ dt = g(v(x)) - g(u(x)).$ so by the chain rule: $h'(x)=v'(x)g'(v(x)) - u'(x)g'(u(x))=v'(x)f(v(x))-u'(x)f(u(x)).$

is there any way to solve the value of integral directly??
• Apr 23rd 2009, 08:02 PM
NonCommAlg
Quote:

Originally Posted by Mathventure
is there any way to solve the value of integral directly??

NO!
• Apr 24th 2009, 05:48 AM
Jester
Quote:

Originally Posted by Mathventure
is there any way to solve the value of integral directly??

If you allow error functions then

$f(x) = \frac{\sqrt{\pi}}{2} \left( \text{erf}(\cos x) - \text{erf}(\sin x)\right)$
• Apr 24th 2009, 07:24 AM
Mathventure
integral problem...
Quote:

Originally Posted by NonCommAlg
NO!

then how can we solve double integral of x*e^(y^2) over region R, where R is region bounded by lines x=0,y=1 and the parabola y=x^2.
• Apr 24th 2009, 07:30 AM
mr fantastic
Quote:

Originally Posted by Mathventure
then how can we solve double integral of x*e^(y^2) over region R, where R is region bounded by lines x=0,y=1 and the parabola y=x^2.

By making sure you integrate in an order that makes the problem tractable:

$\int_{y = 0}^1 \int_{x = 0}^{x = + \sqrt{y}} x e^{y^2} \, dx \, dy$.

There is a world of difference between double integrals and single integrals.

(I can't resist adding that you should have asked danny "How on erf do you do it?" .... Memo to baldeagle: I am the funniest (Rofl))
• Apr 24th 2009, 08:42 AM
Jester
Quote:

Originally Posted by mr fantastic

(I can't resist adding that you should have asked danny "How on erf do you do it?" .... Memo to baldeagle: I am the funniest (Rofl))

Now that's funny (Bigsmile)
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