Jgeurts
Evaluate the following integrals. Show your work.
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zepp
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1/cos^2(x) is sec^2(x) and the integral of that is tan(x)
Jgeurts
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\[\int\limits_{0}^{2}x \sqrt{9-2x^2}dx\]
Jgeurts
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\[\frac{ d }{ dx }\int\limits_{x^2}^{sinx}\sqrt{1+t^2}dt\]
primeralph
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no, just substitute, then everything falls out
Best_Mathematician
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use calculator so easy
Jgeurts
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I need to do it without @best_mathematician
modphysnoob
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integral and derivative cancel each other out
Best_Mathematician
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now u know wht to do right @Jgeurts
Jgeurts
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yes thanks BM
Jgeurts
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@modphysnoob show me please
modphysnoob
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\[\frac{ d }{ dx }\int\limits_{x^2}^{sinx}\sqrt{1+t^2}dt\]
after our supposed integration
F(Sin(x)) - F ( x^2)
take dervaitve
d/dx of F( Sin(x)) = cos(x) f(sin(x)
d/dx of F(x^2) = f (x^2) 2x
go back and plug into your funciton
modphysnoob
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f(t)= Sqrt[1+t^2]
f(sin(x)= sqrt[1+ sin^2(x)]
f(sin(x)*cos(x)= sqrt[1+ sin^2(x)]* cos(x)
modphysnoob
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now the bottom part
f(x^2)2x=sqrt[1+x^2]2x
modphysnoob
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sqrt[1+ sin^2(x)]* cos(x)-sqrt[1+x^2]2x
modphysnoob
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question?
Jgeurts
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Im formulating my question, haha
modphysnoob
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ok
Jgeurts
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so the d/dx means derive the interval?
modphysnoob
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yes, since we are taking a function integrate it and derive , it undoes the integration
Jgeurts
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haha great! i get it, thank you, that one was super hard!
modphysnoob
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I can show you an easy example if you like?
Jgeurts
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@modphysnoob sure that would be great, im learning for my final :)
modphysnoob
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let's to a problem like yours but with easier integrand
|dw:1367723123193:dw|
Jgeurts
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so we would derive sinx to cosx and x2 to 2x
Jgeurts
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then plug in for t and subtract them from each other right?
Jgeurts
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so cosx-2x?
modphysnoob
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so in this case
f(t)= t
when we integrate we will get a function which we will call F(t)
F(t) is evaluated at
F(sin(x)) - F( x^2)
then taken d/dx of
d/dx( F(sin(x)) - F( x^2) )
using chain rule
d/dx ( F(sin(x)) = cos(x) f( sin(x)
modphysnoob
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which is cos(x) sin(x)
Jgeurts
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i see
modphysnoob
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let's do botttom part
d/dx( F (x^2) = 2x f( x^2)
2x x^2= 2x^3
modphysnoob
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so
cos(x) sin(x)-2x^3
modphysnoob
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this is easy integrand so we could actually integrate it and then differntiate it to confirm out result.
Integrate t
t^2/2
plug in limits
(sin^2 x)/2 - x^4/2
take derivative d/dx
cos(x) sin(x)- 2x^3
modphysnoob
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got it?
Jgeurts
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Yes thats great! thank you so much! @modphysnoob