Chapter A6, Problem 48E

To determine

Toevaluate:the given integrals.

Answer to Problem 48E

The required result is

  ${\displaystyle \int {\mathrm{csch}}^2}(5-x)\cdot dx=\coth (5-x)+C$

Explanation of Solution

Given:

  ${\displaystyle \int {\mathrm{csch}}^2}(5-x)\cdot dx$

Calculation:

Assume $u=(5-x)$

Therefore $du=-dx$

Now,

  $\begin{array}{l}{\displaystyle \int {\mathrm{csch}}^2}(5-x)dx=-{\displaystyle \int {\mathrm{csch}}^2}u\cdot du\\ =-(-\coth u)+C\\ =\coth u+C\\ =\coth (5-x)+C\\ {\displaystyle \int {\mathrm{csch}}^2}(5-x)\cdot dx=\coth (5-x)+C\end{array}$

Conclusion:

Therefore, the required result is

  ${\displaystyle \int {\mathrm{csch}}^2}(5-x)\cdot dx=\coth (5-x)+C$