What is the integral of (5x^2+11x+4)/(x^3+3x^2+3x+1) ?

The integral of (5x^2+11x+4)/(x^3+3x^2+3x+1) is

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integral of (5x^2+11x+4)/(x^3+3x^2+3x+1)

\int \frac{5 x ^{2} + 11 x + 4}{x ^{3} + 3 x ^{2} + 3 x + 1} d x

5 (\frac{x ^{3}}{x ^{3} + 3 x ^{2} + 3 x + 1} + ln ∣ x + 1 ∣ + \frac{3}{x + 1} - \frac{3}{2 ( x + 1 ) ^{2}} + \frac{1}{3 ( x + 1 ) ^{3}} + \frac{2 x ^{2}}{3 ( x + 1 ) ^{3}} - \frac{2 ( - 2 x - 1 )}{3 ( x + 1 ) ^{2}}) + 11 (- \frac{x}{2 ( x + 1 ) ^{2}} - \frac{1}{2 ( x + 1 )}) - \frac{2}{( x + 1 ) ^{2}} + C

Solution steps

\int \frac{5 x ^{2} + 11 x + 4}{x ^{3} + 3 x ^{2} + 3 x + 1} d x

Expand

\frac{5 x ^{2} + 11 x + 4}{x ^{3} + 3 x ^{2} + 3 x + 1} : \frac{5 x ^{2}}{x ^{3} + 3 x ^{2} + 3 x + 1} + \frac{11 x}{x ^{3} + 3 x ^{2} + 3 x + 1} + \frac{4}{x ^{3} + 3 x ^{2} + 3 x + 1}

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 ^{2}}{x ^{3} + 3 x ^{2} + 3 x + 1} d x + \int \frac{11 x}{x ^{3} + 3 x ^{2} + 3 x + 1} d x + \int \frac{4}{x ^{3} + 3 x ^{2} + 3 x + 1} d x

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

\int \frac{11 x}{x ^{3} + 3 x ^{2} + 3 x + 1} d x = 11 (- \frac{x}{2 ( x + 1 ) ^{2}} - \frac{1}{2 ( x + 1 )})

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

= 5 (\frac{x ^{3}}{x ^{3} + 3 x ^{2} + 3 x + 1} + ln ∣ x + 1 ∣ + \frac{3}{x + 1} - \frac{3}{2 ( x + 1 ) ^{2}} + \frac{1}{3 ( x + 1 ) ^{3}} + \frac{2 x ^{2}}{3 ( x + 1 ) ^{3}} - \frac{2 ( - 2 x - 1 )}{3 ( x + 1 ) ^{2}}) + 11 (- \frac{x}{2 ( x + 1 ) ^{2}} - \frac{1}{2 ( x + 1 )}) - \frac{2}{( x + 1 ) ^{2}}

Add a constant to the solution

= 5 (\frac{x ^{3}}{x ^{3} + 3 x ^{2} + 3 x + 1} + ln ∣ x + 1 ∣ + \frac{3}{x + 1} - \frac{3}{2 ( x + 1 ) ^{2}} + \frac{1}{3 ( x + 1 ) ^{3}} + \frac{2 x ^{2}}{3 ( x + 1 ) ^{3}} - \frac{2 ( - 2 x - 1 )}{3 ( x + 1 ) ^{2}}) + 11 (- \frac{x}{2 ( x + 1 ) ^{2}} - \frac{1}{2 ( x + 1 )}) - \frac{2}{( x + 1 ) ^{2}} + C

\frac{d}{d x} (\frac{1}{( 4 x + 5 ) ^{2}})

\int \frac{x ln ( x ^{2} + 4 )}{x ^{2} + 4} d x

\int x^{2} x^{3} + 37 d x

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

f (x) = 7 x + 2, at x = 2

What is the integral of (5x^2+11x+4)/(x^3+3x^2+3x+1) ?
The integral of (5x^2+11x+4)/(x^3+3x^2+3x+1) is