Integrate: x^(1/2) / x^(1/2) -x^(1/3)

Answers (1)

Solution: 

                       I=int fracx^frac12x^frac12-x^frac13dx   L.C.M  of 2 and 3 . hence  we put

                          z^6=x,   then  6z^5dz=dx,x^frac12=z^frac62=z^3,

             Rightarrow      I=6int [z^5+z^4+z^3+z^2+z+1+frac1z-1]dz

           Rightarrow      I=6[fracz^66+fracz^55+fracz^44+fracz^32+fracz^22+z+log (z-1)]+c

         Rightarrow         I=6[fracx6+fracx^frac565+fracx^frac234+fracx^frac123+fracx^frac133+x^frac16+log (x^frac16-1)]+c.

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