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

The integral of 1/(x^2+2x+3) is 1/(sqrt(2))arctan((x+1)/(sqrt(2)))+C

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

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

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

Solution steps

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

Complete the square

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

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

Add a constant to the solution

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

\frac{d y}{d x} = \frac{x}{1 + 2 y}

f (x) = \frac{x ^{3}}{3} + 2 x^{2} + 13 x - 5

n = 1 \sum \infty \frac{n}{ln ( 2 n )}

x^{3} - 2 x + 2, at x = 3, y = 21

\frac{\partial}{\partial x} (\frac{8 x + 5 y}{z})

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