What is the integral of (2x)/(x^2+4x+12) ?

The integral of (2x)/(x^2+4x+12) is ln| 1/8 (x+2)^2+1|-sqrt(2)arctan(1/(2sqrt(2))(x+2))+C

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integral of (2x)/(x^2+4x+12)

\int \frac{2 x}{x ^{2} + 4 x + 12} d x

ln \frac{1}{8} (x + 2)^{2} + 1 - 2 arctan (\frac{1}{2 2} (x + 2)) + C

Solution steps

\int \frac{2 x}{x ^{2} + 4 x + 12} d x

Take the constant out:

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

= 2 \cdot \int \frac{x}{x ^{2} + 4 x + 12} d x

Complete the square

x^{2} + 4 x + 12 : (x + 2)^{2} + 8

= 2 \cdot \int \frac{x}{( x + 2 ) ^{2} + 8} d x

Apply u-substitution

: \int \frac{u - 2}{u ^{2} + 8} d u

= 2 \cdot \int \frac{u - 2}{u ^{2} + 8} d u

Apply Integral Substitution

= 2 \cdot \int \frac{2 v - 1}{2 ( v ^{2} + 1 )} 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{2 v - 1}{v ^{2} + 1} d v

Expand

\frac{2 v - 1}{v ^{2} + 1} : \frac{2 v}{v ^{2} + 1} - \frac{1}{v ^{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

= 2 \cdot \frac{1}{2} (\int \frac{2 v}{v ^{2} + 1} d v - \int \frac{1}{v ^{2} + 1} d v)

\int \frac{2 v}{v ^{2} + 1} d v = \frac{1}{2} ln v^{2} + 1

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= 2 \cdot \frac{1}{2} (\frac{1}{2} ln v^{2} + 1 - arctan (v))

Substitute back

= 2 \cdot \frac{1}{2} (\frac{1}{2} ln (\frac{x + 2}{2 2})^{2} + 1 - arctan (\frac{x + 2}{2 2}))

Simplify

2 \cdot \frac{1}{2} (\frac{1}{2} ln (\frac{x + 2}{2 2})^{2} + 1 - arctan (\frac{x + 2}{2 2})) : ln \frac{1}{8} (x + 2)^{2} + 1 - 2 arctan (\frac{1}{2 2} (x + 2))

= ln \frac{1}{8} (x + 2)^{2} + 1 - 2 arctan (\frac{1}{2 2} (x + 2))

Add a constant to the solution

= ln \frac{1}{8} (x + 2)^{2} + 1 - 2 arctan (\frac{1}{2 2} (x + 2)) + C

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\frac{d}{d x} (- 9 cot (x) csc (x))

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What is the integral of (2x)/(x^2+4x+12) ?
The integral of (2x)/(x^2+4x+12) is ln| 1/8 (x+2)^2+1|-sqrt(2)arctan(1/(2sqrt(2))(x+2))+C