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int(x^(7m)+x^(2m)+x^m)(2x^(6m)+7x^m+14)^...

`int(x^(7m)+x^(2m)+x^m)(2x^(6m)+7x^m+14)^(1/m)dx`

Text Solution

Verified by Experts

We have,
`int(x^(3m)+x^(2m)+x^(m))(2x^(2m)+3x^(m)+6)^((1)/(m))dx`, `x gt 0`
`=int(x^(3m-1)+x^(2m-1)+x^(m-1))(2x^(3m)+3x^(2m)+6x^(m))^((1)/(m))dx`
Let us put `2x^(3m)+3x^(2m)+6x^(m)=z`
so that , `6m(x^(3m-1)+x^(2m-1)+x^(m-1))dx=dz`
Thus `intint(x^(3m)+x^(2m)+x^(m))(2x^(2m)+3x^(m)+6)^((1)/(m))dx`
`=intz^((1)/(m))(dz)/(6m)=(1)/(6m)*(z^((1)/(m)+1))/((1)/(m)+1)+C`
`=(1)/(6(m+1))(2x^(3m)+3x^(2m)+6x^(m))^((m+1)/(m))+C`
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