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

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

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

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

2 (\frac{x}{x ^{2} + 4 x + 4} + 2 (- \frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}})) - 5 (\frac{1}{x ^{2} + 4 x + 4} - \frac{3}{2 ( x + 2 ) ^{2}} + \frac{x ^{2} ln ( \frac{1}{2} ∣ x ∣ ) - x ^{2} ln \frac{x}{2} + 1 + 4 x ln ( \frac{1}{2} ∣ x ∣ ) + 2 x - 4 x ln \frac{x}{2} + 1 + 4 ln ( \frac{1}{2} ∣ x ∣ ) - 4 ln \frac{x}{2} + 1 + 6}{4 ( x + 2 ) ^{2}}) + C

Solution steps

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

Expand

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

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

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

\int \frac{5}{x ^{3} + 4 x ^{2} + 4 x} d x = 5 (\frac{1}{x ^{2} + 4 x + 4} - \frac{3}{2 ( x + 2 ) ^{2}} + \frac{x ^{2} ln ( \frac{1}{2} ∣ x ∣ ) - x ^{2} ln \frac{x}{2} + 1 + 4 x ln ( \frac{1}{2} ∣ x ∣ ) + 2 x - 4 x ln \frac{x}{2} + 1 + 4 ln ( \frac{1}{2} ∣ x ∣ ) - 4 ln \frac{x}{2} + 1 + 6}{4 ( x + 2 ) ^{2}})

= 2 (\frac{x}{x ^{2} + 4 x + 4} + 2 (- \frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}})) - 5 (\frac{1}{x ^{2} + 4 x + 4} - \frac{3}{2 ( x + 2 ) ^{2}} + \frac{x ^{2} ln ( \frac{1}{2} ∣ x ∣ ) - x ^{2} ln \frac{x}{2} + 1 + 4 x ln ( \frac{1}{2} ∣ x ∣ ) + 2 x - 4 x ln \frac{x}{2} + 1 + 4 ln ( \frac{1}{2} ∣ x ∣ ) - 4 ln \frac{x}{2} + 1 + 6}{4 ( x + 2 ) ^{2}})

Add a constant to the solution

= 2 (\frac{x}{x ^{2} + 4 x + 4} + 2 (- \frac{1}{x + 2} + \frac{1}{( x + 2 ) ^{2}})) - 5 (\frac{1}{x ^{2} + 4 x + 4} - \frac{3}{2 ( x + 2 ) ^{2}} + \frac{x ^{2} ln ( \frac{1}{2} ∣ x ∣ ) - x ^{2} ln \frac{x}{2} + 1 + 4 x ln ( \frac{1}{2} ∣ x ∣ ) + 2 x - 4 x ln \frac{x}{2} + 1 + 4 ln ( \frac{1}{2} ∣ x ∣ ) - 4 ln \frac{x}{2} + 1 + 6}{4 ( x + 2 ) ^{2}}) + C

\int \frac{1}{\frac{1}{x} + 2} d x

x \to 0 lim (5 x - 6)

2 y = 4 x, y = 4, 2 y + 4 x = 8

\int \frac{d}{1} x^{2} d x

sin (3 x) + cos (8 x)

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