1.

If m is a non-zero number and `int (x^(5m-1)+2x^(4m-1))/(x^(2m)+x^m+1)^3 dx=f(x)+c`, then f(x) is:

Answer» Let `I = int (x^(5m-1)+2x^(4m-1))/(x^(2m)+x^m+1)^3 dx`
`=int (x^(5m-1)+2x^(4m-1))/(x^(6m)(1+x^-m+x^(-2m))^3 dx`
`=int (x^(-m-1)+2x^(-2m-1))/((1+x^-m+x^(-2m))^3 dx`
Let `1+x^-m+x^(-2m) = t`
Then, `-m(x^(-m-1))-2m(x^(-2m-1))dx = dt``=>-m(x^(-m-1)+2x^(-2m-1))dx = dt`
Then,
`I = -1/m int t^-3dt`
`=>I = -1/m (t^-2/(-2))+c`
`=>I = 1/(2m(1+x^-m+x^(-2m))^2) +c`
Comparing it with given equation,
`I = f(x) +c`
`f(x) = 1/(2m(1+x^-m+x^(-2m))^2)`


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