What is int csc x 2 dx

Secans and coscans

Secans and coscans are trigonometric functions.

Secans:f (x) = sec (x)
Cosecant:f (x) = csc (x)

Definition on the unit circle

The functions get their name from the definition in the unit circle. The function values ​​correspond to the length of secant sections:

OT = sec (b) OK = csc (b)

In the right triangle, the secant is the ratio of the hypotenuse to the adjacent side and thus the reciprocal function of the cosine function.

The cosecant is the ratio of the hypotenuse to the opposite cathetus and thus the reciprocal function of the sine function:

sec (α) = lAK lHy = bc csc (α) = lGK lHy = ac
sec (x) = cos (x) 1 csc (x) = sin (x) 1

properties

course

Domain of definition

Secans:−∞
Cosecant:−∞

Range of values

−∞

periodicity

Period length 2⋅π: f (x + 2π) = f (x)

monotony

strictly monotonically decreasing and strictly monotonically increasing sections.

Symmetries

Poles

Secans:x = (n + 21) ⋅π; n∈Z
Cosecant:x = n⋅π; n∈Z

Extreme values

Secans:Minima: x = (2n + 21) ⋅π; n∈ZMaxima: x = (2n − 21) ⋅π; n∈Z
Cosecant:Minima: x = 2n⋅π; n∈ZMaxima: x = (2n − 1) ⋅π; n∈Z

Neither the secant function nor the cosecant function have asymptotes, jumps, turning points or zeros.

Inverse functions

Secans:

On half a period length, e.g. x∈ [0, π] the function is reversible (arc secant):
x = arcsec (y)

cosecant

On half a period length, e.g. x∈ [−2π, 2π] the function is reversible (arccosecans):
x = arccsc (y)

Series development

Secans:

sec (x) = πk = 0∑∞ (2k + 1) 2π2−4x2 (−1) k (8k + 4)

Cosecant:

csc (x) = x1 −2xk = 1∑∞ k2π2 − x2 (−1) k

Derivation

Secans:

dxd sec (x) = sec (x) ⋅tan (x) = csc (x) sec2 (x)

cosecant

dxd csc (x) = - csc (x) ⋅cot (x) = sec (x) csc2 (x)

integral

Secans:

∫sec (x) dx = ln (cos (x) 1 + sin (x))

cosecant

∫csc (x) dx = ln (1 + cos (x) sin (x))

Fear of mathematics is much closer to fear than to awe.

Felix Auerbach

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