What is the integral of x/(x^2+6x+10) ?

The integral of x/(x^2+6x+10) is 1/2 ln|x^2+6x+10|-3arctan(x+3)+C

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integral of x/(x^2+6x+10)

\int \frac{x}{x ^{2} + 6 x + 10} d x

\frac{1}{2} ln x^{2} + 6 x + 10 - 3 arctan (x + 3) + C

Solution steps

\int \frac{x}{x ^{2} + 6 x + 10} d x

Complete the square

x^{2} + 6 x + 10 : (x + 3)^{2} + 1

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

Apply u-substitution

= \int \frac{u - 3}{u ^{2} + 1} d u

Expand

\frac{u - 3}{u ^{2} + 1} : \frac{u}{u ^{2} + 1} - \frac{3}{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{3}{u ^{2} + 1} d u

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

\int \frac{3}{u ^{2} + 1} d u = 3 arctan (u)

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

Substitute back

u = x + 3

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

Simplify

\frac{1}{2} ln (x + 3)^{2} + 1 - 3 arctan (x + 3) : \frac{1}{2} ln x^{2} + 6 x + 10 - 3 arctan (x + 3)

= \frac{1}{2} ln x^{2} + 6 x + 10 - 3 arctan (x + 3)

Add a constant to the solution

= \frac{1}{2} ln x^{2} + 6 x + 10 - 3 arctan (x + 3) + C

What is the integral of x/(x^2+6x+10) ?
The integral of x/(x^2+6x+10) is 1/2 ln|x^2+6x+10|-3arctan(x+3)+C