Integrate [((log(5) x)^2 + sqrt 2*(log(5) x) + 9)/x] with respect to x.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to integrate: [((log(5) x)^2 + sqrt 2*(log(5) x) + 9)/x] with respect to x.

use the relation log(a)x = ln x/ln a

[((log(5) x)^2 + sqrt 2*(log(5) x) + 9)/x]

=> [((ln x/ln 5)^2 + sqrt 2*(ln x/ln 5) + 9)/x]

=> (1/ln 5)(ln x)^2/x + (sqrt 2/ln 5)(ln x)/x + 9/x

Int[(1/ln 5)(ln x)^2/x + (sqrt 2/ln 5)(ln x)/x + 9/x]dx

To find Int[(ln x)^2/x dx]

let y = ln x

dy/dx = 1/x or dy = dx/x

=> Int[y^2 dy]

=> y^3/3

substitute y = ln x

=> (ln x)^3/3

Similarly lnt[ln x/x dx] = (ln x)^2/2

=> (1/ln 5)(ln x)^3/3 + (sqrt 2/ln 5)(ln x)^2/2 + 9 ln x + C

The required integral is (1/ln 5)(ln x)^3/3 + (sqrt 2/ln 5)(ln x)^2/2 + 9 ln x + C

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