Question #fb831

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Answer 1 · 2016-09-06T08:30:31

$= \frac{9}{2} (\sqrt{2} + {sinh}^{- 1} 1)$ $= 9/2 ( sqrt(2) + sinh^(-1) 1 )$

$int_0^3sqrt(x^(2)+9) \ dx$

let $x^2 = 9 sinh^2 z, x = 3 sinh z, dx/dz = 3 cosh z$

$implies int_(sinh^(-1) 0)^(sinh^(-1) 1) sqrt(9 sinh^2 z+9) \ 3 cosh z \ dz$

$implies 9int_(sinh^(-1) 0)^(sinh^(-1) 1) cosh^2 z \ dz$

Using
$cosh 2x =cosh ^2 x+ sinh ^{2} x=2 sinh ^2 x+1 = 2 cosh^2 x-1$

$implies 9/2 int_(sinh^(-1) 0)^(sinh^(-1) 1) cosh 2z + 1\ dz$

$9/2 [ 1/2sinh 2z + z ]_(sinh^(-1) 0)^(sinh^(-1) 1)$

Using $sinh 2x = 2 sinh x cosh x$

$= 9/2 [ sinh z cosh z + z ]_(sinh^(-1) 0)^(sinh^(-1) 1)$

$= 9/2 [ sinh z sqrt(1 + sinh^2 z) + z ]_(sinh^(-1) 0)^(sinh^(-1) 1)$

$= 9/2 ( 1 sqrt(1 + 1^2 ) + sinh^(-1) 1 )$

$= 9/2 ( sqrt(2) + sinh^(-1) 1 )$