Solveeit Logo
Login

Question

Question: The value of the integral \(\int_{- 1/2}^{1/2}\sqrt{\left( \frac{x + 1}{x - 1} \right)^{2} + \left( ...

The value of the integral ∫−1/21/2(x+1x−1)2+(x−1x+1)2−2\int_{- 1/2}^{1/2}\sqrt{\left( \frac{x + 1}{x - 1} \right)^{2} + \left( \frac{x - 1}{x + 1} \right)^{2} - 2}dx is -

A

2 log 43\frac{4}{3}

B

4 log 43\frac{4}{3}

C

log 43\frac{4}{3}

D

None of these

Answer

4 log 43\frac{4}{3}

Explanation

Solution

∫−1/21/2(x+1x−1−x−1x+1)2\int_{- 1/2}^{1/2}\sqrt{\left( \frac{x + 1}{x - 1} - \frac{x - 1}{x + 1} \right)^{2}}

= ∫−1/21/2∣x+1x−1−x−1x+1∣dx\int_{- 1/2}^{1/2}{\left| \frac{x + 1}{x - 1} - \frac{x - 1}{x + 1} \right|dx}

= ∫−1/21/2∣4xx2−1∣dx=2∫01/2∣4xx2−1∣\int_{- 1/2}^{1/2}{\left| \frac{4x}{x^{2} - 1} \right|dx = 2\int_{0}^{1/2}\left| \frac{4x}{x^{2} - 1} \right|}dx

(Q integrand is an even function)

= –2∫01/2(4xx2−1)dx\int_{0}^{1/2}{\left( \frac{4x}{x^{2} - 1} \right)dx}

(∵4xx2−1<0inthei芀te(0,12))()\left( \because\frac{4x}{x^{2} - 1} < 0inthei芀te\left( 0,\frac{1}{2} \right) \right)()

= –4[log (1 – x2) ]01/2\rbrack_{0}^{1/2} = – 4 (log⁡34)\left( \log\frac{3}{4} \right) = 4 log (43)\left( \frac{4}{3} \right).