What is the integral of 3/(2x^2+x-10) ?

The integral of 3/(2x^2+x-10) is -1/3 (ln| 10/9+4/9 x|-ln|-8/9+4/9 x|)+C

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integral of 3/(2x^2+x-10)

\int \frac{3}{2 x ^{2} + x - 10} d x

- \frac{1}{3} (ln \frac{10}{9} + \frac{4}{9} x - ln - \frac{8}{9} + \frac{4}{9} x) + C

Solution steps

\int \frac{3}{2 x ^{2} + x - 10} d x

Take the constant out:

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

= 3 \cdot \int \frac{1}{2 x ^{2} + x - 10} d x

Complete the square

2 x^{2} + x - 10 : 2 (x + \frac{1}{4})^{2} - \frac{81}{8}

= 3 \cdot \int \frac{1}{2 ( x + \frac{1}{4} ) ^{2} - \frac{81}{8}} d x

Apply u-substitution

: \int \frac{8}{16 u ^{2} - 81} d u

= 3 \cdot \int \frac{8}{16 u ^{2} - 81} d u

Take the constant out:

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

= 3 \cdot 8 \cdot \int \frac{1}{16 u ^{2} - 81} d u

Apply Integral Substitution

= 3 \cdot 8 \cdot \int \frac{1}{36 ( v ^{2} - 1 )} d v

Take the constant out:

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

= 3 \cdot 8 \cdot \frac{1}{36} \cdot \int \frac{1}{v ^{2} - 1} d v

Factor

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

= 3 \cdot 8 \cdot \frac{1}{36} \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

= 3 \cdot 8 \cdot \frac{1}{36} (- \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}

= 3 \cdot 8 \cdot \frac{1}{36} (- (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}))

Substitute back

= 3 \cdot 8 \cdot \frac{1}{36} (- (\frac{ln \frac{4}{9} ( x + \frac{1}{4} ) + 1}{2} - \frac{ln \frac{4}{9} ( x + \frac{1}{4} ) - 1}{2}))

Simplify

3 \cdot 8 \cdot \frac{1}{36} (- (\frac{ln \frac{4}{9} ( x + \frac{1}{4} ) + 1}{2} - \frac{ln \frac{4}{9} ( x + \frac{1}{4} ) - 1}{2})) : - \frac{1}{3} (ln \frac{10}{9} + \frac{4}{9} x - ln - \frac{8}{9} + \frac{4}{9} x)

= - \frac{1}{3} (ln \frac{10}{9} + \frac{4}{9} x - ln - \frac{8}{9} + \frac{4}{9} x)

Add a constant to the solution

= - \frac{1}{3} (ln \frac{10}{9} + \frac{4}{9} x - ln - \frac{8}{9} + \frac{4}{9} x) + C

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What is the integral of 3/(2x^2+x-10) ?
The integral of 3/(2x^2+x-10) is -1/3 (ln| 10/9+4/9 x|-ln|-8/9+4/9 x|)+C