Definite Integral

Calculus Level 3

∫ 1 2 1 ( x + 1 ) x 2 − 1 d x = 1 a \large \int \limits _{ 1 } ^{ 2 } \frac{ 1 } { ( x + 1 ) \sqrt{ x^{ 2 } - 1 } } \, dx = \frac{ 1 } {\sqrt{ a } }

Find a a .


The answer is 3.

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Hamza A by Hamza A , 19, USA

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2 solutions

∫ 1 2 d x ( x + 1 x − 1 ) 3 2 . ( x − 1 ) 2 p u t x + 1 x − 1 = t d x ( x − 1 ) 2 = d t − 2 ∫ ∞ 3 d t − 2 ( t ) 3 2 = [ 1 t ] 3 ∞ = 1 3 \int _{ 1 }^{ 2 }{ \frac { dx }{ { \left( \frac { x+1 }{ x-1 } \right) }^{ \frac { 3 }{ 2 } }.{ \left( x-1 \right) }^{ 2 } } } \\ put\quad \frac { x+1 }{ x-1 } =t\quad \quad \quad \frac { dx }{ { \left( x-1 \right) }^{ 2 } } =\frac { dt }{ -2 } \\ \int _{ \infty }^{ 3 }{ \frac { dt }{ -2{ \left( t \right) }^{ \frac { 3 }{ 2 } } } } =\left[ \frac { 1 }{ \sqrt { t } } \right] \begin{matrix} 3 \\ \infty \end{matrix}=\frac { 1 }{ \sqrt { 3 } }

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Moderator note:

When doing a change of variables, you should also ensure that the function is differentiable in the region that you are using.

You should add words to explain what you are doing.

Rudraksh Shukla
Jan 18, 2016

Substitute x=sec(t) and solve.

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