INTEGRATION_Trigonometric Functions and Hyperbolic Functions

INTEGRATION: Trigonometric Functions & Hyperbolic Functions

BACKGROUND THEORY
Corresponding to each trigonometric (or hyperbolic) differentiation formula is an integration formula.
Although these are always given in a formula sheet, it will help you a great deal in the exam in terms of
saving time if you learn the formulas by heart.

Trigonometric Functions Hyperbolic Functions
d d
dusin u  cos u du sinh u  cosh u
dx dx
dx dx

cos u du  sin u  C 
cosh u du  sinh u  C
d du d
dx
 cos u  sin u du cosh u  sinh u
dx
dx dx

sin u du  cos u  C 
sinh u du  cosh u  C
d d
du tan u  sec u dutanh u  sech u
2 2

dx dx
dx dx

sec u du  tan u  C
2

sech u du  tanh u  C
2


d d
dusec u  sec u  tan u dusech u  sech u  tanh u 
dx dx
dx dx

sec u tan u du  sec u  C 
sech u tanh u du  sech u  C
d d
ducsc u  csc u cot u ducsch u  csch u coth u 
dx dx
dx dx

csc u cot u du  csc u  C 
csch u coth u du  csch u  C
d du d du
dx
cot u  csc2 u
dx
 coth u  csch2 u

dx dx

csc u du  cot u  C
2

csch u du  coth u  C
2




Special Standard Integrals:
Trigonometric Functions Hyperbolic Functions



 tan udu  ln sec u  C  tanh u du  ln  cosh u   C
 cot u du  ln sin u  C  coth u du  ln sinh u  C
 sec u du  ln sec u  tan u  C  sech u du  arctan sinh u   C

,  csc u du  ln csc u  cot u  C  csch u du  ln tanh  u
2  C

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