What is the integral of 7/(x^2+x-12) ?

The integral of 7/(x^2+x-12) is -ln| 8/7+2/7 x|+ln|-6/7+2/7 x|+C

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integral of 7/(x^2+x-12)

\int \frac{7}{x ^{2} + x - 12} d x

- ln \frac{8}{7} + \frac{2}{7} x + ln - \frac{6}{7} + \frac{2}{7} x + C

Solution steps

\int \frac{7}{x ^{2} + x - 12} d x

Take the constant out:

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

= 7 \cdot \int \frac{1}{x ^{2} + x - 12} d x

Complete the square

x^{2} + x - 12 : (x + \frac{1}{2})^{2} - \frac{49}{4}

= 7 \cdot \int \frac{1}{( x + \frac{1}{2} ) ^{2} - \frac{49}{4}} d x

Apply u-substitution

: \int \frac{4}{4 u ^{2} - 49} d u

= 7 \cdot \int \frac{4}{4 u ^{2} - 49} d u

Take the constant out:

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

= 7 \cdot 4 \cdot \int \frac{1}{4 u ^{2} - 49} d u

Apply Integral Substitution

= 7 \cdot 4 \cdot \int \frac{1}{14 ( v ^{2} - 1 )} d v

Take the constant out:

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

= 7 \cdot 4 \cdot \frac{1}{14} \cdot \int \frac{1}{v ^{2} - 1} d v

Factor

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

= 7 \cdot 4 \cdot \frac{1}{14} \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

= 7 \cdot 4 \cdot \frac{1}{14} (- \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}

= 7 \cdot 4 \cdot \frac{1}{14} (- (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}))

Substitute back

= 7 \cdot 4 \cdot \frac{1}{14} (- (\frac{ln \frac{2}{7} ( x + \frac{1}{2} ) + 1}{2} - \frac{ln \frac{2}{7} ( x + \frac{1}{2} ) - 1}{2}))

Simplify

7 \cdot 4 \cdot \frac{1}{14} (- (\frac{ln \frac{2}{7} ( x + \frac{1}{2} ) + 1}{2} - \frac{ln \frac{2}{7} ( x + \frac{1}{2} ) - 1}{2})) : - ln \frac{8}{7} + \frac{2}{7} x + ln - \frac{6}{7} + \frac{2}{7} x

= - ln \frac{8}{7} + \frac{2}{7} x + ln - \frac{6}{7} + \frac{2}{7} x

Add a constant to the solution

= - ln \frac{8}{7} + \frac{2}{7} x + ln - \frac{6}{7} + \frac{2}{7} x + C

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What is the integral of 7/(x^2+x-12) ?
The integral of 7/(x^2+x-12) is -ln| 8/7+2/7 x|+ln|-6/7+2/7 x|+C