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The Integral of square root (x+square root (x^2+2 )

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Solution    We have , 

                  I=int sqrtx+sqrtx^2+2dx 

        Let          x+sqrtx^2+2=t

  Rightarrow                sqrtx^2+2=(t-x)

Rightarrow                  x^2+2=t^2+x^2-2tx

Rightarrow                  x=frac12(t-frac2t)Rightarrow dx=frac12(1+frac4t^2)dt

herefore                  I= frac12int sqrtt(1+frac4t^2)dt=frac12int (t^frac12+4t^frac-32)dt

Rightarrow                 I=frac12[fract^frac32frac32+4fract^frac-12frac-12]+c=fract^frac323-4t^frac-12+c

Rightarrow                I=frac13(sqrtx+sqrtx^2+2)^frac32-4(sqrtx+sqrtx^2+2)^frac-12+c 

       

 

 

 

 

 

 

Posted by

Deependra Verma

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