What is the integral of 1/(sqrt(x^2-4x+1)) ?

The integral of 1/(sqrt(x^2-4x+1)) is ln|sqrt(1/3 (x^2-4x+1))+1/(sqrt(3))(x-2)|+C

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integral of 1/(sqrt(x^2-4x+1))

\int \frac{1}{x ^{2} - 4 x + 1} d x

ln \frac{1}{3} (x^{2} - 4 x + 1) + \frac{1}{3} (x - 2) + C

Solution steps

\int \frac{1}{x ^{2} - 4 x + 1} d x

Complete the square

x^{2} - 4 x + 1 : (x - 2)^{2} - 3

= \int \frac{1}{( x - 2 ) ^{2} - 3} d x

Apply u-substitution

= \int \frac{1}{u ^{2} - 3} d u

Apply Trigonometric Substitution

= \int sec (v) d v

Use the common integral:

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= ln ∣ tan (v) + sec (v) ∣

Substitute back

= ln tan (arcsec (\frac{1}{3} (x - 2))) + sec (arcsec (\frac{1}{3} (x - 2)))

Simplify

ln tan (arcsec (\frac{1}{3} (x - 2))) + sec (arcsec (\frac{1}{3} (x - 2))) : ln \frac{1}{3} (x^{2} - 4 x + 1) + \frac{1}{3} (x - 2)

= ln \frac{1}{3} (x^{2} - 4 x + 1) + \frac{1}{3} (x - 2)

Add a constant to the solution

= ln \frac{1}{3} (x^{2} - 4 x + 1) + \frac{1}{3} (x - 2) + C

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What is the integral of 1/(sqrt(x^2-4x+1)) ?
The integral of 1/(sqrt(x^2-4x+1)) is ln|sqrt(1/3 (x^2-4x+1))+1/(sqrt(3))(x-2)|+C