What is the integral of (x+2)/(x^2+11x+18) ?

The integral of (x+2)/(x^2+11x+18) is ln|2x+18|+C

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integral of (x+2)/(x^2+11x+18)

\int \frac{x + 2}{x ^{2} + 11 x + 18} d x

ln ∣ 2 x + 18 ∣ + C

Solution steps

\int \frac{x + 2}{x ^{2} + 11 x + 18} d x

Complete the square

x^{2} + 11 x + 18 : (x + \frac{11}{2})^{2} - \frac{49}{4}

= \int \frac{x + 2}{( x + \frac{11}{2} ) ^{2} - \frac{49}{4}} d x

Apply u-substitution

: \int \frac{2}{2 u + 7} d u

= \int \frac{2}{2 u + 7} d u

Take the constant out:

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

= 2 \cdot \int \frac{1}{2 u + 7} d u

Apply u-substitution

: \int \frac{1}{2 v} d v

= 2 \cdot \int \frac{1}{2 v} d v

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 2 \cdot \frac{1}{2} ln ∣ v ∣

Substitute back

= 2 \cdot \frac{1}{2} ln 2 (x + \frac{11}{2}) + 7

Simplify

2 \cdot \frac{1}{2} ln 2 (x + \frac{11}{2}) + 7 : ln ∣ 2 x + 18 ∣

= ln ∣ 2 x + 18 ∣

Add a constant to the solution

= ln ∣ 2 x + 18 ∣ + C

\int_{0}^{y} 8 d x

f (x) = (3 x - 5)^{\frac{1}{2}}, at x = 2

\int \frac{1}{( 25 x ^{2} + 16 ) ^{2}} d x

\frac{d}{d x} (- x^{- 2} - 1)

(x^{2} + 4 \cdot x + 2)^{9}

What is the integral of (x+2)/(x^2+11x+18) ?
The integral of (x+2)/(x^2+11x+18) is ln|2x+18|+C