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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

\int - \frac{1}{x + 1} d x = - ln ∣ x + 1 ∣

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

Simplify

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

Add a constant to the solution

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

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