Integrate: (x^3m +x^2m +x^m)(2x^2m+3x^m+6) ^1/m

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Solution:   

       \ \ I=int (x^3m+x^2m+x^m)(2x^2m+3x^m+6)^frac1mdx\ \I=int (x^3m+x^2m+x^m)frac(2x^3m+3x^2m+6x^m)^frac1mxdx\ \ I=int (x^3m-1+x^2m-1+x^m-1)(2x^3m+3x^2m+6x^m)^frac1m dx\ \ Put hspace0.2cm2x^3m+3x^2m+6x^m=tRightarrow 6m(x^3m-1+x^2m-1+x^m-1)dx=dt\ \ I=int t^frac1mfracdt6m=frac16mfract^(frac1m)+1(frac1m+1)+c=frac16(m+1)(2x^3m+3x^2m+6x^m)fracm+1m+C.

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