What is the integral of (2x+1)/(x^2+11x+30) ?

The integral of (2x+1)/(x^2+11x+30) is ln|4x^2+44x+120|+10ln|2x+12|-10ln|2x+10|+C

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integral of (2x+1)/(x^2+11x+30)

integral

\frac{2 x + 1}{x ^{2} + 11 x + 30}

ln 4 x^{2} + 44 x + 120 + 10 ln ∣ 2 x + 12 ∣ - 10 ln ∣ 2 x + 10 ∣ + C

Solution steps

\int \frac{2 x + 1}{x ^{2} + 11 x + 30} d x

Complete the square

x^{2} + 11 x + 30 : (x + \frac{11}{2})^{2} - \frac{1}{4}

= \int \frac{2 x + 1}{( x + \frac{11}{2} ) ^{2} - \frac{1}{4}} d x

Apply u-substitution

: \int \frac{4 ( 2 u - 10 )}{4 u ^{2} - 1} d u

= \int \frac{4 ( 2 u - 10 )}{4 u ^{2} - 1} d u

Take the constant out:

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

= 4 \cdot \int \frac{2 u - 10}{4 u ^{2} - 1} d u

Expand

\frac{2 u - 10}{4 u ^{2} - 1} : \frac{2 u}{4 u ^{2} - 1} - \frac{10}{4 u ^{2} - 1}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 4 (\int \frac{2 u}{4 u ^{2} - 1} d u - \int \frac{10}{4 u ^{2} - 1} d u)

\int \frac{2 u}{4 u ^{2} - 1} d u = \frac{1}{4} ln 4 u^{2} - 1

\int \frac{10}{4 u ^{2} - 1} d u = - 5 (\frac{1}{2} ln ∣ 2 u + 1 ∣ - \frac{1}{2} ln ∣ 2 u - 1 ∣)

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

Substitute back

u = x + \frac{11}{2}

= 4 (\frac{1}{4} ln 4 (x + \frac{11}{2})^{2} - 1 - (- 5 (\frac{1}{2} ln 2 (x + \frac{11}{2}) + 1 - \frac{1}{2} ln 2 (x + \frac{11}{2}) - 1)))

Simplify

4 (\frac{1}{4} ln 4 (x + \frac{11}{2})^{2} - 1 - (- 5 (\frac{1}{2} ln 2 (x + \frac{11}{2}) + 1 - \frac{1}{2} ln 2 (x + \frac{11}{2}) - 1))) : ln 4 x^{2} + 44 x + 120 + 10 ln ∣ 2 x + 12 ∣ - 10 ln ∣ 2 x + 10 ∣

= ln 4 x^{2} + 44 x + 120 + 10 ln ∣ 2 x + 12 ∣ - 10 ln ∣ 2 x + 10 ∣

Add a constant to the solution

= ln 4 x^{2} + 44 x + 120 + 10 ln ∣ 2 x + 12 ∣ - 10 ln ∣ 2 x + 10 ∣ + C

\frac{( u - 1 ) ^{0.5}}{u}

\frac{( y + 1 ) ^{2}}{y}

(1 - y)^{4}

(2 x + 1)^{- 3}

\frac{1}{( 9 x ^{2} + 4 )}

What is the integral of (2x+1)/(x^2+11x+30) ?
The integral of (2x+1)/(x^2+11x+30) is ln|4x^2+44x+120|+10ln|2x+12|-10ln|2x+10|+C