integral(x*ln(sqrt(1+x^2)) 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 (1+x^2)= t, so that, if we'll differentiate it, the result will be:

(2*x*dx/2sqrt (1+x^2) )=dt

x*dx/sqrt (1+x^2) =dt

x*dx=sqrt (1+x^2) *dt, but sqrt (1+x^2)= t


Now, we'll write the integral depending on the variable "t":

integral(x*ln(sqrt(1+x^2)) dx)=integral(ln t*tdt) 

Now, we can use the integration by parts method:

Integral (f' * g)=f*g-Integral(f*g')

We'll choose "ln t" as being f function:

f=ln t, so that f'=1/t

g'=t dt, so that g=Integral (t) dt=t^2/2

integral(ln t*tdt) =(t^2/2)*ln t-integral[ (1/t)*t^2/2]

integral(ln t*tdt) =(t^2/2)*ln t-(1/2)*(t^2/2) + C

But sqrt (1+x^2)= t, so

integral(x*ln(sqrt(1+x^2)) dx)= ((1+x^2)/2)*ln sqrt (1+x^2)-1/4*(1+x^2) +C