Differentiate and integrate hyprbolic functions

Differentiate and integrate hyprbolic functions

Differentiating Hyperbolic Functions

• The derivatives of the hyperbolic functions are stated below:
  • The derivative of sinh x with respect to x, denoted as d/dx(sinh x), is cosh x.
  • The derivative of cosh x with respect to x, denoted as d/dx(cosh x), is sinh x.
  • The derivative of tanh x with respect to x, denoted as d/dx(tanh x), is sech^2 x.
  • The derivative of coth x with respect to x, denoted as d/dx(coth x), is -csch^2 x.
  • The derivative of sech x with respect to x, denoted as d/dx(sech x), is -sech x tanh x.
  • The derivative of csch x with respect to x, denoted as d/dx(csch x), is -csch x coth x.

Integrating Hyperbolic Functions

• The integration of hyperbolic functions uses the inverse formula of their derivatives:
  • The integral of cosh x dx, denoted ∫cosh x dx, gives sinh x + C.
  • The integral of sinh x dx, denoted ∫sinh x dx, gives cosh x + C.
  • The integral of sech^2 x dx, denoted ∫sech^2 x dx, gives tanh x + C.
  • The integral of csch^2 x dx, denoted ∫csch^2 x dx, gives -coth x + C.
  • The integral of sech x tanh x dx, denoted ∫sech x tanh x dx, gives -sech x + C.
  • The integral of csch x coth x dx, denoted ∫csch x coth x dx, gives -csch x + C.

Integrating Functions involving Hyperbolic Functions

• Certain other integration problems involving hyperbolic functions can be solved by substitution or other integration techniques. Be familiar with the methods to integrate functions like ∫x sinh x dx or ∫x cosh x dx. The techniques used when integrating these functions are comparable to those used when integrating standard trigonometric functions.
• The constant ‘C’ represents the constant of integration, which captures an arbitrary constant added to the end of all indefinite integrals. This is linked to the fact that the derivative of any constant is zero.