What is the integral from 6 to 8 of 1/(x^2-1) ?

The integral from 6 to 8 of 1/(x^2-1) is -(2ln(3)-2ln(7)+ln(5))/2

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integral from 6 to 8 of 1/(x^2-1)

\int_{6}^{8} \frac{1}{x ^{2} - 1} d x

- \frac{2 ln ( 3 ) - 2 ln ( 7 ) + ln ( 5 )}{2}

Decimal Notation

0.04257 \dots

Solution steps

\int_{6}^{8} \frac{1}{x ^{2} - 1} d x

Factor

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

= \int_{6}^{8} \frac{1}{- ( - x ^{2} + 1 )} d x

Take the constant out:

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

= - \int_{6}^{8} \frac{1}{- x ^{2} + 1} d x

Use the common integral:

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

= - [\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2}]_{6}^{8}

Simplify

- [\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2}]_{6}^{8} : - [\frac{1}{2} (ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣)]_{6}^{8}

= - [\frac{1}{2} (ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣)]_{6}^{8}

Compute the boundaries:

\frac{2 ln ( 3 ) - 2 ln ( 7 ) + ln ( 5 )}{2}

= - \frac{2 ln ( 3 ) - 2 ln ( 7 ) + ln ( 5 )}{2}

x \to 0 lim (x sec (x) csc (x))

\frac{d y}{d x} + y + 3 sin (x) = 0

\int_{0}^{\frac{π}{3}} (2 sec^{2} (x)) d x

y = \frac{7}{x}, (4, \frac{7}{2})

\frac{d}{d x} (50 - x^{2} - x)

What is the integral from 6 to 8 of 1/(x^2-1) ?
The integral from 6 to 8 of 1/(x^2-1) is -(2ln(3)-2ln(7)+ln(5))/2