Verify the integration formula.
-$$\int x \operatorname { csch } ^ { - 1 } x d x = \frac { x ^ { 2 } } { 2 } \operatorname { csch } ^ { - 1 } x + \frac { 1 } { 2 } \sqrt { 1 + x ^ { 2 } } + C$$

Question

Quizplus · Accepted Answer

The integral on the left-hand side can be evaluated using the substitution $u = \operatorname { csch } ^ { - 1 } x$, which gives $du = - \frac { 1 } { x \sqrt { 1 + x ^ { 2 } } } dx$ and $x = \sinh u$. Substituting these into the integral, we get$$\int x \operatorname { csch } ^ { - 1 } x d x = \int \sinh u \ u \ du = \frac { 1 } { 2 } \sinh ^ { 2 } u + C = \frac { 1 } { 2 } x ^ { 2 } \operatorname { csch } ^ { - 1 } x + \frac { 1 } { 2 } \sqrt { 1 + x ^ { 2 } } + C$$which is the right-hand side of the given formula.

Verify the Integration Formula