`int x/sqrt(x^2-6x+5)` Complete the square and find the indefinite integral

Expert Answers

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Let's complete the square of the denominator of the integrand,


Now apply integral substitution:`u=x-3`



Now apply the sum rule,


Now let's evaluate the first integral ,


Apply integral substitution:`v=u^2-4`



Take the constant out and apply the power rule,




Substitute back `v=u^2-4`


Now let's evaluate the second integral,


Take the constant out,


Use the standard integral:`int1/sqrt(x^2-a^2)dx=ln|x+sqrt(x^2-a^2)|`

`=3ln|u+sqrt(u^2-2^2)|` ,

So add the result of the two integrals,


Substitute back `u=x-3` and add a constant C to the solution,





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