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integral(cos(sqrt（x）) dx) first make subsitution then use integration by parts to...
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First of all, you have to make the substitution:
sqrt x= t, so that, if we'll differentiate it, the result will be:
From this expression, we'll find out dx, which is:
dx=(2sqrt x)dt, but sqrt x=t, so
Now, we'll write the integral depending on the variable "t":
integral(cos(sqrt x) dx)=integral[(cost)(2tdt)]
Now, we can use the integration by parts method:
Integral (f' * g)=f*g-Integral(f*g')
We'll choose "2t" as being f function:
f=2t, so that f'=2
g'=cos t dt, so that g=Integral (cos t) dt=sin t
Integral[(cost)(2tdt)]=2t*sin t - Integral (2sint)dt
Integral[(cost)(2tdt)]=2t*sin t -2(-cos t)
Integral[(cost)(2tdt)]=2t*sin t+2cos t
Integral[(cost)(2tdt)]=2*(t*sin t + cos t)
But sqrt x= t, so
2*(t*sin t + cos t)=2*(sqrt x*sin (sqrt x) + cos (sqrt x))
Posted by giorgiana1976 on February 12, 2010 at 2:46 AM (Answer #1)
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