Integration of Hyperbolic Functions: A Comprehensive Guide with Examples and Exercises, Slides of Differential and Integral Calculus

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Integration of Hyperbolic Functions

TOPIC

OBJECTIVES

• (^) identify the different hyperbolic functions;
• (^) find the integral of given hyperbolic functions;
• (^) determine the difference between the integrals

of hyperbolic functions; and

• (^) evaluate integrals involving hyperbolic

functions.

Differentiation Formulas

1. d^ ^ sinh u ^  cosh udu
2. d  cosh u   sinh udu d^ ^ u ^ h udu 2
3. tanh  sec d  u  h udu 2

4. coth   csc

5. d  sec hu   sec hu tanh udu
6. d  cschu  cschucothudu

. Note : The hyperbolic functions are defined in terms of the exponential functions. Its differentials may also be found by differentiating its equivalent exponential form. Similarly, the integrals of the hyperbolic functions can be derived by integrating the exponential form equivalent.

Hyperbolic Functions Trigonometric

Functions

cosh x sinh x 1 2 2   1 tanh x sech x 2 2   coth x 1 csch x 2 2   sinh( xy)sinhxcoshycoshxsinh y cosh  xy coshxcoshysinhxsinhy   1 tanhxtanhy tanhx tanh y tanh x y  

  ^ ^

1 tanxtany tanx tan y tan x y     cos  xy  cosxcosysinxsiny sin  x  y sinxcosy  cosxsiny

x x

2 2

1  tan  sec

cos sin 1 2 2 x  x 

cot x 1 csc x

2 2

Identities: Hyperbolic Functions vs. Trigonometric Functions

Hyperbolic

Functions

Trigonometric

Functions

Identities: Hyperbolic Functions vs. Trigonometric Functions sinh 2x = 2 sinh x cosh x sinh x  cosh 2 x 1  / 2 2   cosh x  cosh 2 x 1  / 2 2   x cosh x  sinhx  e x cosh x sinhx e   

cos x  1 cos 2 x / 2

2   sin x  1 cos 2 x / 2 2   cos 2x = cos 2x – sin2x sin 2x = 2sinx cosx cosh 2x = cosh 2x +sinh2x

Example: Evaluate the following integrals:

1. sinh 1  3 x dx

2. e coshe dx 2 x 2 x

dy cosh y a

2 tanh y

ln 3 0 2

4. sech tdt

5. sinh xcosh xdx

3 2

 csch xcoth xdx x 1

4 2

CLASSWORK 

1. sinh 4 xcosh 4 xdx 
2. xcsch 2 x dx 2 2