What is the integral of (-5x-3)/(x^2+3x+4) ?

The integral of (-5x-3)/(x^2+3x+4) is -5/2 ln|4x^2+12x+16|+9/(sqrt(7))arctan(1/(sqrt(7))(2x+3))+C

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integral of (-5x-3)/(x^2+3x+4)

\int \frac{- 5 x - 3}{x ^{2} + 3 x + 4} d x

- \frac{5}{2} ln 4 x^{2} + 12 x + 16 + \frac{9}{7} arctan (\frac{1}{7} (2 x + 3)) + C

Solution steps

\int \frac{- 5 x - 3}{x ^{2} + 3 x + 4} d x

Expand

\frac{- 5 x - 3}{x ^{2} + 3 x + 4} : - \frac{5 x}{x ^{2} + 3 x + 4} - \frac{3}{x ^{2} + 3 x + 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{5 x}{x ^{2} + 3 x + 4} d x - \int \frac{3}{x ^{2} + 3 x + 4} d x

\int \frac{5 x}{x ^{2} + 3 x + 4} d x = 10 (\frac{1}{4} ln 4 x^{2} + 12 x + 16 - \frac{3}{2 7} arctan (\frac{2 x + 3}{7}))

\int \frac{3}{x ^{2} + 3 x + 4} d x = \frac{6}{7} arctan (\frac{2 x + 3}{7})

= - 10 (\frac{1}{4} ln 4 x^{2} + 12 x + 16 - \frac{3}{2 7} arctan (\frac{2 x + 3}{7})) - \frac{6}{7} arctan (\frac{2 x + 3}{7})

Simplify

- 10 (\frac{1}{4} ln 4 x^{2} + 12 x + 16 - \frac{3}{2 7} arctan (\frac{2 x + 3}{7})) - \frac{6}{7} arctan (\frac{2 x + 3}{7}) : - \frac{5}{2} ln 4 x^{2} + 12 x + 16 + \frac{9}{7} arctan (\frac{1}{7} (2 x + 3))

= - \frac{5}{2} ln 4 x^{2} + 12 x + 16 + \frac{9}{7} arctan (\frac{1}{7} (2 x + 3))

Add a constant to the solution

= - \frac{5}{2} ln 4 x^{2} + 12 x + 16 + \frac{9}{7} arctan (\frac{1}{7} (2 x + 3)) + C

\frac{\partial}{\partial x} (x e^{w})

x \to 90 lim (tan (x))

\int \frac{e ^{x}}{x ^{4}} d x

y^{^{'}} - 5 y = 2 x^{2} - 5 x + 8

\frac{d}{d x} (- x^{3} + 20 x^{2} + 21 x)

What is the integral of (-5x-3)/(x^2+3x+4) ?
The integral of (-5x-3)/(x^2+3x+4) is -5/2 ln|4x^2+12x+16|+9/(sqrt(7))arctan(1/(sqrt(7))(2x+3))+C