What is the integral of x/(x^2+6x+11) ?

The integral of x/(x^2+6x+11) is 1/2 ln|x^2+6x+11|-3/(sqrt(2))arctan(1/(sqrt(2))(x+3))+C

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integral of x/(x^2+6x+11)

\int \frac{x}{x ^{2} + 6 x + 11} d x

\frac{1}{2} ln x^{2} + 6 x + 11 - \frac{3}{2} arctan (\frac{1}{2} (x + 3)) + C

Solution steps

\int \frac{x}{x ^{2} + 6 x + 11} d x

Complete the square

x^{2} + 6 x + 11 : (x + 3)^{2} + 2

= \int \frac{x}{( x + 3 ) ^{2} + 2} d x

Apply u-substitution

: \int \frac{u - 3}{u ^{2} + 2} d u

= \int \frac{u - 3}{u ^{2} + 2} d u

Expand

\frac{u - 3}{u ^{2} + 2} : \frac{u}{u ^{2} + 2} - \frac{3}{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{3}{u ^{2} + 2} d u

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

\int \frac{3}{u ^{2} + 2} d u = \frac{3}{2} arctan (\frac{u}{2})

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

Substitute back

u = x + 3

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

Simplify

\frac{1}{2} ln (x + 3)^{2} + 2 - \frac{3}{2} arctan (\frac{x + 3}{2}) : \frac{1}{2} ln x^{2} + 6 x + 11 - \frac{3}{2} arctan (\frac{1}{2} (x + 3))

= \frac{1}{2} ln x^{2} + 6 x + 11 - \frac{3}{2} arctan (\frac{1}{2} (x + 3))

Add a constant to the solution

= \frac{1}{2} ln x^{2} + 6 x + 11 - \frac{3}{2} arctan (\frac{1}{2} (x + 3)) + C

\frac{d}{d x} ((5 - 3 x)^{3})

e^{6 e^{x}}

\int_{0}^{1} (2 x - 2 x^{2}) d x

\frac{d}{d x} (x^{4} + x^{3} + x^{2} + 1)

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

What is the integral of x/(x^2+6x+11) ?
The integral of x/(x^2+6x+11) is 1/2 ln|x^2+6x+11|-3/(sqrt(2))arctan(1/(sqrt(2))(x+3))+C