What is the integral from-1/2 to 0 of (2x+4)/(x^2-1) ?

The integral from-1/2 to 0 of (2x+4)/(x^2-1) is -ln(3/4)-2ln(3)

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integral from-1/2 to 0 of (2x+4)/(x^2-1)

\int_{- \frac{1}{2}}^{0} \frac{2 x + 4}{x ^{2} - 1} d x

- ln (\frac{3}{4}) - 2 ln (3)

Decimal

- 1.90954 \dots

Solution steps

\int_{- \frac{1}{2}}^{0} \frac{2 x + 4}{x ^{2} - 1} d x

Expand

\frac{2 x + 4}{x ^{2} - 1} : \frac{2 x}{x ^{2} - 1} + \frac{4}{x ^{2} - 1}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{- \frac{1}{2}}^{0} \frac{2 x}{x ^{2} - 1} d x + \int_{- \frac{1}{2}}^{0} \frac{4}{x ^{2} - 1} d x

\int_{- \frac{1}{2}}^{0} \frac{2 x}{x ^{2} - 1} d x = - ln (\frac{3}{4})

\int_{- \frac{1}{2}}^{0} \frac{4}{x ^{2} - 1} d x = - 2 ln (3)

= - ln (\frac{3}{4}) - 2 ln (3)

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What is the integral from-1/2 to 0 of (2x+4)/(x^2-1) ?
The integral from-1/2 to 0 of (2x+4)/(x^2-1) is -ln(3/4)-2ln(3)