What is the integral of (12)/(4x^2+5x+1) ?

The integral of (12)/(4x^2+5x+1) is -8(1/2 ln| 8/3+8/3 x|-1/2 ln| 2/3+8/3 x|)+C

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integral of (12)/(4x^2+5x+1)

\int \frac{12}{4 x ^{2} + 5 x + 1} d x

- 8 (\frac{1}{2} ln \frac{8}{3} + \frac{8}{3} x - \frac{1}{2} ln \frac{2}{3} + \frac{8}{3} x) + C

Solution steps

\int \frac{12}{4 x ^{2} + 5 x + 1} d x

Take the constant out:

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

= 12 \cdot \int \frac{1}{4 x ^{2} + 5 x + 1} d x

Complete the square

4 x^{2} + 5 x + 1 : 4 (x + \frac{5}{8})^{2} - \frac{9}{16}

= 12 \cdot \int \frac{1}{4 ( x + \frac{5}{8} ) ^{2} - \frac{9}{16}} d x

Apply u-substitution

: \int \frac{16}{64 u ^{2} - 9} d u

= 12 \cdot \int \frac{16}{64 u ^{2} - 9} d u

Take the constant out:

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

= 12 \cdot 16 \cdot \int \frac{1}{64 u ^{2} - 9} d u

Apply Integral Substitution

= 12 \cdot 16 \cdot \int \frac{1}{24 ( v ^{2} - 1 )} d v

Take the constant out:

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

= 12 \cdot 16 \cdot \frac{1}{24} \cdot \int \frac{1}{v ^{2} - 1} d v

Factor

v^{2} - 1 : - (- v^{2} + 1)

= 12 \cdot 16 \cdot \frac{1}{24} \cdot \int \frac{1}{- ( - v ^{2} + 1 )} d v

Take the constant out:

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

= 12 \cdot 16 \cdot \frac{1}{24} (- \int \frac{1}{- v ^{2} + 1} d v)

Use the common integral:

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

= 12 \cdot 16 \cdot \frac{1}{24} (- (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}))

Substitute back

= 12 \cdot 16 \cdot \frac{1}{24} (- (\frac{ln \frac{8}{3} ( x + \frac{5}{8} ) + 1}{2} - \frac{ln \frac{8}{3} ( x + \frac{5}{8} ) - 1}{2}))

Simplify

12 \cdot 16 \cdot \frac{1}{24} (- (\frac{ln \frac{8}{3} ( x + \frac{5}{8} ) + 1}{2} - \frac{ln \frac{8}{3} ( x + \frac{5}{8} ) - 1}{2})) : - 8 (\frac{1}{2} ln \frac{8}{3} + \frac{8}{3} x - \frac{1}{2} ln \frac{2}{3} + \frac{8}{3} x)

= - 8 (\frac{1}{2} ln \frac{8}{3} + \frac{8}{3} x - \frac{1}{2} ln \frac{2}{3} + \frac{8}{3} x)

Add a constant to the solution

= - 8 (\frac{1}{2} ln \frac{8}{3} + \frac{8}{3} x - \frac{1}{2} ln \frac{2}{3} + \frac{8}{3} x) + C

\int (x + 3)^{2} (3 - x)^{7} d x

x \to 9 lim (x)

f (x) = x^{2} - x - 2

\frac{d}{d x} ((6 x^{2} - 12 x + 12) e^{x})

\frac{6 x ^{2} + 4 x + 4}{x}

What is the integral of (12)/(4x^2+5x+1) ?
The integral of (12)/(4x^2+5x+1) is -8(1/2 ln| 8/3+8/3 x|-1/2 ln| 2/3+8/3 x|)+C