integral(cos(sqrt(x)) dx) first make subsitution then use integration by parts to evaluate integral. show steps

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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:

(1/2sqrt x)dx=dt

From this expression, we'll find out dx, which is:

dx=dt/(1/2sqrt x)

dx=(2sqrt x)dt, but sqrt x=t, so

dx=2t dt

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))

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