What is the integral of (x^2+2x+3)/(x^3+3x^2+9x) ?

The integral of (x^2+2x+3)/(x^3+3x^2+9x) is 1/3 ln|x^3+3x^2+9x|+C

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integral of (x^2+2x+3)/(x^3+3x^2+9x)

\int \frac{x ^{2} + 2 x + 3}{x ^{3} + 3 x ^{2} + 9 x} d x

\frac{1}{3} ln x^{3} + 3 x^{2} + 9 x + C

Solution steps

\int \frac{x ^{2} + 2 x + 3}{x ^{3} + 3 x ^{2} + 9 x} d x

Apply u-substitution

: \int \frac{1}{3 u} d u

= \int \frac{1}{3 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3} ln ∣ u ∣

Substitute back

u = x^{3} + 3 x^{2} + 9 x

= \frac{1}{3} ln x^{3} + 3 x^{2} + 9 x

Add a constant to the solution

= \frac{1}{3} ln x^{3} + 3 x^{2} + 9 x + C

y = x^{2} - 2 x + 3, [- 2, 2]

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What is the integral of (x^2+2x+3)/(x^3+3x^2+9x) ?
The integral of (x^2+2x+3)/(x^3+3x^2+9x) is 1/3 ln|x^3+3x^2+9x|+C