What is the integral of (x+10)/(x^2+6x+13) ?

The integral of (x+10)/(x^2+6x+13) is 1/2 ln|x^2+6x+13|+7/2 arctan((x+3)/2)+C

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integral of (x+10)/(x^2+6x+13)

\int \frac{x + 10}{x ^{2} + 6 x + 13} d x

\frac{1}{2} ln x^{2} + 6 x + 13 + \frac{7}{2} arctan (\frac{x + 3}{2}) + C

Solution steps

\int \frac{x + 10}{x ^{2} + 6 x + 13} d x

Apply u-substitution

: \int \frac{u}{u ^{2} - 14 u + 53} d u

= \int \frac{u}{u ^{2} - 14 u + 53} d u

Complete the square

u^{2} - 14 u + 53 : (u - 7)^{2} + 4

= \int \frac{u}{( u - 7 ) ^{2} + 4} d u

Apply u-substitution

: \int \frac{v + 7}{v ^{2} + 4} d v

= \int \frac{v + 7}{v ^{2} + 4} d v

Expand

\frac{v + 7}{v ^{2} + 4} : \frac{v}{v ^{2} + 4} + \frac{7}{v ^{2} + 4}

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{v}{v ^{2} + 4} d v + \int \frac{7}{v ^{2} + 4} d v

\int \frac{v}{v ^{2} + 4} d v = \frac{1}{2} ln v^{2} + 4

\int \frac{7}{v ^{2} + 4} d v = \frac{7}{2} arctan (\frac{v}{2})

= \frac{1}{2} ln v^{2} + 4 + \frac{7}{2} arctan (\frac{v}{2})

Substitute back

= \frac{1}{2} ln (x + 10 - 7)^{2} + 4 + \frac{7}{2} arctan (\frac{x + 10 - 7}{2})

Simplify

\frac{1}{2} ln (x + 10 - 7)^{2} + 4 + \frac{7}{2} arctan (\frac{x + 10 - 7}{2}) : \frac{1}{2} ln x^{2} + 6 x + 13 + \frac{7}{2} arctan (\frac{x + 3}{2})

= \frac{1}{2} ln x^{2} + 6 x + 13 + \frac{7}{2} arctan (\frac{x + 3}{2})

Add a constant to the solution

= \frac{1}{2} ln x^{2} + 6 x + 13 + \frac{7}{2} arctan (\frac{x + 3}{2}) + C

\frac{\partial}{\partial x} (- x + y)

\int x^{2} tan^{3} (x^{3}) d x

(x^{2} + 4) y^{^{'}} + 3 x y = x, y (0) = 1

9 x^{4} - 9 x^{2}, 8 x^{2}

What is the integral of (x+10)/(x^2+6x+13) ?
The integral of (x+10)/(x^2+6x+13) is 1/2 ln|x^2+6x+13|+7/2 arctan((x+3)/2)+C