Solve \int _( 1/( sqrt(3)) )^( sqrt(3)) 8/(1-x^(2)) d

Evaluate

\int_{\frac{1}{3}}^{3} \frac{8}{1 - x ^{2}} d x

Use function domain and discontinuity points to transform the expression with the formula \int_{a}^{c} f (x) d x = \int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x

\int_{\frac{1}{3}}^{1} \frac{8}{1 - x ^{2}} d x + \int_{1}^{3} \frac{8}{1 - x ^{2}} d x

By definition,rewrite the improper integral using one-sided limit and a definite integral

a \to 1^{-} lim (\int_{\frac{1}{3}}^{a} \frac{8}{1 - x ^{2}} d x) + \int_{1}^{3} \frac{8}{1 - x ^{2}} d x

By definition,rewrite the improper integral using one-sided limit and a definite integral

a \to 1^{-} lim (\int_{\frac{1}{3}}^{a} \frac{8}{1 - x ^{2}} d x) + a \to 1^{+} lim (\int_{a}^{3} \frac{8}{1 - x ^{2}} d x)

Evaluate the integral

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Evaluate

\int_{\frac{1}{3}}^{a} \frac{8}{1 - x ^{2}} d x

Evaluate the integral

\int \frac{8}{1 - x ^{2}} d x

Rewrite the expression

\int 8 \times \frac{1}{1 - x ^{2}} d x

Use the property of integral \int k f (x) d x = k \int f (x) d x

8 \times \int \frac{1}{1 - x ^{2}} d x

Use the property of integral \int \frac{1}{a ^{2} - x ^{2}} d x = \frac{1}{2 a} ln \frac{x + a}{x - a}

8 \times \frac{1}{2} ln (\frac{x + 1}{x - 1})

Multiply the terms

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4 ln (\frac{x + 1}{x - 1})

Return the limits

(4 ln (\frac{x + 1}{x - 1}))_{\frac{1}{3}}^{a}

Substitute the values into formula

4 ln (\frac{a + 1}{a - 1}) - 4 ln \frac{\frac{1}{3} + 1}{\frac{1}{3} - 1}

Add the numbers

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4 ln (\frac{a + 1}{a - 1}) - 4 ln \frac{\frac{3 + 3}{3}}{\frac{1}{3} - 1}

Subtract the numbers

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4 ln (\frac{a + 1}{a - 1}) - 4 ln \frac{\frac{3 + 3}{3}}{\frac{3 - 3}{3}}

Calculate the absolute value

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4 ln (\frac{a + 1}{a - 1}) - 4 ln (\frac{3 + 3}{3 - 3})

Rewrite the expression

ln (\frac{a + 1}{a - 1}) \times 4 - ln (\frac{3 + 3}{3 - 3}) \times 4

Factor the expression

(ln (\frac{a + 1}{a - 1}) - ln (\frac{3 + 3}{3 - 3})) \times 4

Subtract the terms

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ln \frac{( 3 - 3 ) ∣ a + 1 ∣}{( 3 + 3 ) ∣ a - 1 ∣} \times 4

Multiply the terms

4 ln \frac{( 3 - 3 ) ∣ a + 1 ∣}{( 3 + 3 ) ∣ a - 1 ∣}

a \to 1^{-} lim 4 ln \frac{( 3 - 3 ) ∣ a + 1 ∣}{( 3 + 3 ) ∣ a - 1 ∣} + a \to 1^{+} lim (\int_{a}^{3} \frac{8}{1 - x ^{2}} d x)

Evaluate the integral

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a \to 1^{-} lim 4 ln \frac{( 3 - 3 ) ∣ a + 1 ∣}{( 3 + 3 ) ∣ a - 1 ∣} + a \to 1^{+} lim 4 ln \frac{( 3 + 1 ) ∣ a - 1 ∣}{( 3 - 1 ) ∣ a + 1 ∣}

Evaluate the limit

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(+ \infty) + a \to 1^{+} lim 4 ln \frac{( 3 + 1 ) ∣ a - 1 ∣}{( 3 - 1 ) ∣ a + 1 ∣}

Evaluate the limit

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(+ \infty) + (- \infty)

Solution

Diverges