What is the integral of (10x+44)/(x^2+6x+58) ?

The integral of (10x+44)/(x^2+6x+58) is 5ln|x^2+6x+58|+2arctan(1/7 (x+3))+C

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integral of (10x+44)/(x^2+6x+58)

\int \frac{10 x + 44}{x ^{2} + 6 x + 58} d x

5 ln x^{2} + 6 x + 58 + 2 arctan (\frac{1}{7} (x + 3)) + C

Solution steps

\int \frac{10 x + 44}{x ^{2} + 6 x + 58} d x

Complete the square

x^{2} + 6 x + 58 : (x + 3)^{2} + 49

= \int \frac{10 x + 44}{( x + 3 ) ^{2} + 49} d x

Apply u-substitution

: \int \frac{10 u + 14}{u ^{2} + 49} d u

= \int \frac{10 u + 14}{u ^{2} + 49} d u

Expand

\frac{10 u + 14}{u ^{2} + 49} : \frac{10 u}{u ^{2} + 49} + \frac{14}{u ^{2} + 49}

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{10 u}{u ^{2} + 49} d u + \int \frac{14}{u ^{2} + 49} d u

\int \frac{10 u}{u ^{2} + 49} d u = 5 ln u^{2} + 49

\int \frac{14}{u ^{2} + 49} d u = 2 arctan (\frac{u}{7})

= 5 ln u^{2} + 49 + 2 arctan (\frac{u}{7})

Substitute back

u = x + 3

= 5 ln (x + 3)^{2} + 49 + 2 arctan (\frac{x + 3}{7})

Simplify

5 ln (x + 3)^{2} + 49 + 2 arctan (\frac{x + 3}{7}) : 5 ln x^{2} + 6 x + 58 + 2 arctan (\frac{1}{7} (x + 3))

= 5 ln x^{2} + 6 x + 58 + 2 arctan (\frac{1}{7} (x + 3))

Add a constant to the solution

= 5 ln x^{2} + 6 x + 58 + 2 arctan (\frac{1}{7} (x + 3)) + C

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What is the integral of (10x+44)/(x^2+6x+58) ?
The integral of (10x+44)/(x^2+6x+58) is 5ln|x^2+6x+58|+2arctan(1/7 (x+3))+C