What is the integral of (30)/(x^2+x-6) ?

The integral of (30)/(x^2+x-6) is -12(1/2 ln| 6/5+2/5 x|-1/2 ln|-4/5+2/5 x|)+C

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integral of (30)/(x^2+x-6)

\int \frac{30}{x ^{2} + x - 6} d x

- 12 (\frac{1}{2} ln \frac{6}{5} + \frac{2}{5} x - \frac{1}{2} ln - \frac{4}{5} + \frac{2}{5} x) + C

Solution steps

\int \frac{30}{x ^{2} + x - 6} d x

Take the constant out:

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

= 30 \cdot \int \frac{1}{x ^{2} + x - 6} d x

Complete the square

x^{2} + x - 6 : (x + \frac{1}{2})^{2} - \frac{25}{4}

= 30 \cdot \int \frac{1}{( x + \frac{1}{2} ) ^{2} - \frac{25}{4}} d x

Apply u-substitution

: \int \frac{4}{4 u ^{2} - 25} d u

= 30 \cdot \int \frac{4}{4 u ^{2} - 25} d u

Take the constant out:

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

= 30 \cdot 4 \cdot \int \frac{1}{4 u ^{2} - 25} d u

Apply Integral Substitution

= 30 \cdot 4 \cdot \int \frac{1}{10 ( v ^{2} - 1 )} d v

Take the constant out:

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

= 30 \cdot 4 \cdot \frac{1}{10} \cdot \int \frac{1}{v ^{2} - 1} d v

Factor

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

= 30 \cdot 4 \cdot \frac{1}{10} \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

= 30 \cdot 4 \cdot \frac{1}{10} (- \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}

= 30 \cdot 4 \cdot \frac{1}{10} (- (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}))

Substitute back

= 30 \cdot 4 \cdot \frac{1}{10} (- (\frac{ln \frac{2}{5} ( x + \frac{1}{2} ) + 1}{2} - \frac{ln \frac{2}{5} ( x + \frac{1}{2} ) - 1}{2}))

Simplify

30 \cdot 4 \cdot \frac{1}{10} (- (\frac{ln \frac{2}{5} ( x + \frac{1}{2} ) + 1}{2} - \frac{ln \frac{2}{5} ( x + \frac{1}{2} ) - 1}{2})) : - 12 (\frac{1}{2} ln \frac{6}{5} + \frac{2}{5} x - \frac{1}{2} ln - \frac{4}{5} + \frac{2}{5} x)

= - 12 (\frac{1}{2} ln \frac{6}{5} + \frac{2}{5} x - \frac{1}{2} ln - \frac{4}{5} + \frac{2}{5} x)

Add a constant to the solution

= - 12 (\frac{1}{2} ln \frac{6}{5} + \frac{2}{5} x - \frac{1}{2} ln - \frac{4}{5} + \frac{2}{5} x) + C

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What is the integral of (30)/(x^2+x-6) ?
The integral of (30)/(x^2+x-6) is -12(1/2 ln| 6/5+2/5 x|-1/2 ln|-4/5+2/5 x|)+C