What is the integral of (3x-2)/(x^2+8x+25) ?

The integral of (3x-2)/(x^2+8x+25) is 3/2 ln|x^2+8x+25|-14/3 arctan(1/3 (x+4))+C

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integral of (3x-2)/(x^2+8x+25)

\int \frac{3 x - 2}{x ^{2} + 8 x + 25} d x

\frac{3}{2} ln x^{2} + 8 x + 25 - \frac{14}{3} arctan (\frac{1}{3} (x + 4)) + C

Solution steps

\int \frac{3 x - 2}{x ^{2} + 8 x + 25} d x

Expand

\frac{3 x - 2}{x ^{2} + 8 x + 25} : \frac{3 x}{x ^{2} + 8 x + 25} - \frac{2}{x ^{2} + 8 x + 25}

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{3 x}{x ^{2} + 8 x + 25} d x - \int \frac{2}{x ^{2} + 8 x + 25} d x

\int \frac{3 x}{x ^{2} + 8 x + 25} d x = \frac{3}{2} ln x^{2} + 8 x + 25 - 4 arctan (\frac{1}{3} (x + 4))

\int \frac{2}{x ^{2} + 8 x + 25} d x = \frac{2}{3} arctan (\frac{1}{3} (x + 4))

= \frac{3}{2} ln x^{2} + 8 x + 25 - 4 arctan (\frac{1}{3} (x + 4)) - \frac{2}{3} arctan (\frac{1}{3} (x + 4))

Add similar elements:

- 4 arctan (\frac{1}{3} (x + 4)) - \frac{2}{3} arctan (\frac{1}{3} (x + 4)) = - \frac{14}{3} arctan (\frac{1}{3} (x + 4))

= \frac{3}{2} ln x^{2} + 8 x + 25 - \frac{14}{3} arctan (\frac{1}{3} (x + 4))

Add a constant to the solution

= \frac{3}{2} ln x^{2} + 8 x + 25 - \frac{14}{3} arctan (\frac{1}{3} (x + 4)) + C

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What is the integral of (3x-2)/(x^2+8x+25) ?
The integral of (3x-2)/(x^2+8x+25) is 3/2 ln|x^2+8x+25|-14/3 arctan(1/3 (x+4))+C