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Question: \(\int_{}^{}\frac{1}{(x + 1)\sqrt{x^{2}–1}}\)dx is equal to –...

∫1(x+1)x2–1\int_{}^{}\frac{1}{(x + 1)\sqrt{x^{2}–1}}dx is equal to –

A

12\frac { 1 } { 2 } x–1x+1\sqrt{\frac{x–1}{x + 1}}+ C

B

x+1x–1\sqrt{\frac{x + 1}{x–1}} + C

C

12\frac { 1 } { 2 } x+1x–1\sqrt{\frac{x + 1}{x–1}}+c

D

x–1x+1\sqrt{\frac{x–1}{x + 1}} + C

Answer

x–1x+1\sqrt{\frac{x–1}{x + 1}} + C

Explanation

Solution

Let x + 1 = 1t\frac { 1 } { \mathrm { t } } then dx = – dt

I = × (−1t2)\left( - \frac { 1 } { t ^ { 2 } } \right) dt = – ∫dt1−2t\int \frac { \mathrm { dt } } { \sqrt { 1 - 2 \mathrm { t } } }

= –dt = –+C =+C