What is the integral of x/(x^2+4x+5) ?

The integral of x/(x^2+4x+5) is 1/2 ln|x^2+4x+5|-2arctan(x+2)+C

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integral of x/(x^2+4x+5)

\int \frac{x}{x ^{2} + 4 x + 5} d x

\frac{1}{2} ln x^{2} + 4 x + 5 - 2 arctan (x + 2) + C

Solution steps

\int \frac{x}{x ^{2} + 4 x + 5} d x

Complete the square

x^{2} + 4 x + 5 : (x + 2)^{2} + 1

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

Apply u-substitution

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

Expand

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

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

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

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

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

Substitute back

u = x + 2

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

Simplify

\frac{1}{2} ln (x + 2)^{2} + 1 - 2 arctan (x + 2) : \frac{1}{2} ln x^{2} + 4 x + 5 - 2 arctan (x + 2)

= \frac{1}{2} ln x^{2} + 4 x + 5 - 2 arctan (x + 2)

Add a constant to the solution

= \frac{1}{2} ln x^{2} + 4 x + 5 - 2 arctan (x + 2) + C

What is the integral of x/(x^2+4x+5) ?
The integral of x/(x^2+4x+5) is 1/2 ln|x^2+4x+5|-2arctan(x+2)+C