Mathematics

# Solve $\int { \dfrac { 1 }{ \sqrt { { a }^{ 2 }+{ \left( x-c \right) }^{ 2 } } } }$

##### SOLUTION
$put \ x=c+atan\Theta$
${dx}=asec^{2}\Theta{d\Theta}$
$Now \ the \ inegration \ reduces \ to:$
$\int{\dfrac {asec^{2}\Theta{d\Theta}}{asec\Theta}}$
$=\int{sec\Theta{d\Theta}}$
$=ln(sec\Theta+tan\Theta) +C\to(1)$
$tan\Theta=\dfrac{x-c}{a} \ and \ sec\Theta=\dfrac{\sqrt{a^{2}+(x-c)^{2}}}{a}$
$putting \ the \ values \ of \ sec\Theta \ and \ tan\Theta \ in (1)$
$required \ answer \ will \ be \ ln(\dfrac{x-c}{a}+\dfrac{\sqrt{a^{2}+(x-c)^{2}}}{a})+C$
$or;$
$ln((x-c)+\sqrt{a^{2}+(x-c)^{2}})+C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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