What is the integral from 0 to 2 of 1/(x^2-x-2) ?

The integral from 0 to 2 of 1/(x^2-x-2) is diverges

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integral from 0 to 2 of 1/(x^2-x-2)

\int_{0}^{2} \frac{1}{x ^{2} - x - 2} d x

diverges

Solution steps

\int_{0}^{2} \frac{1}{x ^{2} - x - 2} d x

Complete the square

x^{2} - x - 2 : (x - \frac{1}{2})^{2} - \frac{9}{4}

= \int_{0}^{2} \frac{1}{( x - \frac{1}{2} ) ^{2} - \frac{9}{4}} d x

Apply u-substitution

: \int_{- \frac{1}{2}}^{\frac{3}{2}} \frac{4}{4 u ^{2} - 9} d u

= \int_{- \frac{1}{2}}^{\frac{3}{2}} \frac{4}{4 u ^{2} - 9} d u

Take the constant out:

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

= 4 \cdot \int_{- \frac{1}{2}}^{\frac{3}{2}} \frac{1}{4 u ^{2} - 9} d u

Apply Integral Substitution

= 4 \cdot \int_{- \frac{1}{3}}^{1} \frac{1}{6 ( v ^{2} - 1 )} d v

Take the constant out:

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

= 4 \cdot \frac{1}{6} \cdot \int_{- \frac{1}{3}}^{1} \frac{1}{v ^{2} - 1} d v

Factor

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

= 4 \cdot \frac{1}{6} \cdot \int_{- \frac{1}{3}}^{1} \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

= 4 \cdot \frac{1}{6} (- \int_{- \frac{1}{3}}^{1} \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}

= 4 \cdot \frac{1}{6} (- [\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}]_{- \frac{1}{3}}^{1})

Simplify

4 \cdot \frac{1}{6} (- [\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}]_{- \frac{1}{3}}^{1}) : - \frac{2}{3} [\frac{ln ∣ v + 1 ∣ - ln ∣ v - 1 ∣}{2}]_{- \frac{1}{3}}^{1}

= - \frac{2}{3} [\frac{ln ∣ v + 1 ∣ - ln ∣ v - 1 ∣}{2}]_{- \frac{1}{3}}^{1}

Compute the boundaries:

diverges

= diverges

(x^{3} + 5) e^{x}

\frac{d}{d x} (in y (x))

\int_{1}^{10} \frac{1}{x ^{2} - 4 x - 32} d x

g (x) = \frac{1}{5 x ^{2} + 4 x}

\frac{ln ( x )}{3 x}, \frac{( ln ( x ) ) ^{2}}{3 x}

What is the integral from 0 to 2 of 1/(x^2-x-2) ?
The integral from 0 to 2 of 1/(x^2-x-2) is diverges