Evaluate the integral in terms of an inverse hyperbolic function.
-$$\int _ { \sqrt { 5 } } ^ { 6 \sqrt { 5 } } \frac { d x } { \sqrt { 36 + x ^ { 2 } } }$$
A) $\sinh ^ { - 1 } ( \sqrt { 5 } ) - \sinh ^ { - 1 } \left( \frac { \sqrt { 5 } } { 6 } \right)$
B) $- \frac { 1 } { 6 } \operatorname { csch } ^ { - 1 } ( \sqrt { 5 } ) - \frac { 1 } { 6 } \operatorname { csch } ^ { - 1 } \left( \frac { \sqrt { 5 } } { 6 } \right)$
C) $\cosh ^ { - 1 } ( \sqrt { 5 } ) - \cosh ^ { - 1 } \left( \frac { \sqrt { 5 } } { 6 } \right)$
D) $- \frac { 1 } { 6 } \operatorname { sech } ^ { - 1 } ( \sqrt { 5 } ) + \frac { 1 } { 6 } \operatorname { sech } ^ { - 1 } \left( \frac { \sqrt { 5 } } { 6 } \right)$

Question

Quizplus · Accepted Answer

The integral is of the form $\int \frac { d x } { \sqrt { a ^ { 2 } + x ^ { 2 } } } = \sinh ^ { - 1 } \left( \frac { x } { a } \right) + C$, where $a$ is a constant. In this case, $a = 6$, so the integral is $\sinh ^ { - 1 } \left( \frac { x } { 6 } \right) + C$. Evaluating the integral from $\sqrt { 5 }$ to $6 \sqrt { 5 }$ gives $\sinh ^ { - 1 } \left( \frac { 6 \sqrt { 5 } } { 6 } \right) - \sinh ^ { - 1 } \left( \frac { \sqrt { 5 } } { 6 } \right) = \sinh ^ { - 1 } ( \sqrt { 5 } ) - \sinh ^ { - 1 } \left( \frac { \sqrt { 5 } } { 6 } \right)$.

Evaluate the Integral in Terms of an Inverse Hyperbolic Function ∫565dx36+x2\int _ { \sqrt { 5 } } ^ { 6 \sqrt { 5 } } \frac { d x } { \sqrt { 36 + x ^ { 2 } } }∫5​65​​36+x2​dx​

Evaluate the Integral in Terms of an Inverse Hyperbolic Function $\int _ { \sqrt { 5 } } ^ { 6 \sqrt { 5 } } \frac { d x } { \sqrt { 36 + x ^ { 2 } } }$