What is the integral of 1/(sqrt(3-2x+x^2)) ?

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

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integral of 1/(sqrt(3-2x+x^2))

\int \frac{1}{3 - 2 x + x ^{2}} d x

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

Solution steps

\int \frac{1}{3 - 2 x + x ^{2}} d x

Complete the square

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

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

Apply u-substitution

= \int \frac{1}{u ^{2} + 2} 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 (arctan (\frac{1}{2} (x - 1))) + sec (arctan (\frac{1}{2} (x - 1)))

Simplify

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

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

Add a constant to the solution

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

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