What is the integral of 1/(x^2-4x+6) ?

The integral of 1/(x^2-4x+6) is 1/(sqrt(2))arctan((x-2)/(sqrt(2)))+C

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integral of 1/(x^2-4x+6)

\int \frac{1}{x ^{2} - 4 x + 6} d x

\frac{1}{2} arctan (\frac{x - 2}{2}) + C

Solution steps

\int \frac{1}{x ^{2} - 4 x + 6} d x

Complete the square

x^{2} - 4 x + 6 : (x - 2)^{2} + 2

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

Apply u-substitution

: \int \frac{1}{u ^{2} + 2} d u

= \int \frac{1}{u ^{2} + 2} d u

Apply Integral Substitution

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1}{v ^{2} + 1} d v

Use the common integral:

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= \frac{1}{2} arctan (v)

Substitute back

= \frac{1}{2} arctan (\frac{x - 2}{2})

Add a constant to the solution

= \frac{1}{2} arctan (\frac{x - 2}{2}) + C

\frac{d y}{d x}, x^{2} + y^{2} = (2 x^{2} + 4 y^{2} - x) 2

\frac{d y}{d x} = 2 (2 x + y)^{2}

\int 2 cos (2 x - 5) d x

x \to 42 lim (\frac{x ^{2}}{7} - \frac{6}{x})

\frac{d}{d x} ((ln (2 x))^{3})

What is the integral of 1/(x^2-4x+6) ?
The integral of 1/(x^2-4x+6) is 1/(sqrt(2))arctan((x-2)/(sqrt(2)))+C