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

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

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

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

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

Solution steps

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

Complete the square

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

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

Apply u-substitution

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

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

Expand

\frac{u - 1}{u ^{2} + 2} : \frac{u}{u ^{2} + 2} - \frac{1}{u ^{2} + 2}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

= \frac{1}{2} ln u^{2} + 2 - \frac{1}{2} arctan (\frac{u}{2})

Substitute back

u = x + 1

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

Simplify

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

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

Add a constant to the solution

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

\int \frac{x ^{2} + 4 x + 27}{( x - 2 ) ( x ^{2} + 9 )} d x

\frac{1}{x ^{3}}

\int x 2^{- x^{2}} d x

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

\int \frac{e ^{a r c t a n (x)}}{( 1 + x ^{2} ) ^{\frac{3}{2}}} d x

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