Calculus
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Boundless Calculus
Inverse Functions and Advanced Integration
Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions
Calculus Textbooks Boundless Calculus Inverse Functions and Advanced Integration Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions
Calculus Textbooks Boundless Calculus Inverse Functions and Advanced Integration
Calculus Textbooks Boundless Calculus
Calculus Textbooks
Calculus
Concept Version 12
Created by Boundless

Inverse Trigonometric Functions: Differentiation and Integration

It is useful to know the derivatives and antiderivatives of the inverse trigonometric functions.

Learning Objective

  • Practice integrating and differentiating inverse trigonometric functions


Key Points

    • The inverse trigonometric functions "undo" the trigonometric functions sin\sinsin, cos\coscos, and tan\tantan.
    • The inverse trigonometric functions are arcsin\arcsinarcsin, arccos\arccosarccos, and arctan\arctanarctan.
    • Memorizing their derivatives and antiderivatives can be useful.

Term

  • trigonometric

    relating to the functions used in trigonometry: sin\sinsin, cos\coscos, tan\tantan, csc\csccsc, cot\cotcot, sec\secsec


Full Text

The inverse trigonometric functions are also known as the "arc functions". There are three common notations for inverse trigonometric functions. The arcsine function, for instance, could be written as sin−1\sin^{-1}sin​−1​​, asin\text{asin}asin, or, as is used on this page, arcsin\arcsinarcsin. Inverse trigonometric functions include arcsin\arcsinarcsin and arccos\arccosarccos; arctan\arctanarctan and arccot\text{arccot}arccot; and arcsec\text{arcsec}arcsec and arccsc\text{arccsc}arccsc. They can be thought of as the inverses of the corresponding trigonometric functions.

Arcsine and Arccosine

The usual principal values of the arcsin(x)\arcsin(x)arcsin(x) and arccos(x)\arccos(x)arccos(x) functions graphed on the Cartesian plane.

Arctangent and Arccotangent

The usual principal values of the arctan(x)\text{arctan}(x)arctan(x) and arccot(x)\text{arccot}(x)arccot(x) functions graphed on the Cartesian plane.

Arcsecant and Arccosecant

Principal values of the arcsec(x)\text{arcsec}(x)arcsec(x) and arccsc(x)\text{arccsc}(x)arccsc(x) functions graphed on the Cartesian plane.

The differentiation of trigonometric functions is the mathematical process of finding the rate at which a trigonometric function changes with respect to a variable.

The integration of trigonometric functions involves finding the antiderivative.

The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse trigonometric functions.

∫arcsec(ax)dx=x arcsec(ax)−1aarctanh(1−1a2x2+C)\displaystyle{\int \text{arcsec}( a x) dx = x \ \text{arcsec}(a x) - \dfrac{1}{a} \text{arctanh}\left(\sqrt{1-\dfrac{1}{a^2 x^2}} + C\right)}∫arcsec(ax)dx=x arcsec(ax)−​a​​1​​arctanh(√​1−​a​2​​x​2​​​​1​​​​​+C)

∫arccsc(ax)dx=x arccsc(ax)+1aarctanh(1−1a2x2+C)\displaystyle{\int \text{arccsc}( a x) dx = x \ \text{arccsc}(a x) + \dfrac{1}{a} \text{arctanh}\left(\sqrt{1-\dfrac{1}{a^2 x^2}} + C\right)}∫arccsc(ax)dx=x arccsc(ax)+​a​​1​​arctanh(√​1−​a​2​​x​2​​​​1​​​​​+C)

CCC is used for the arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus each function has an infinite number of antiderivatives.

The derivatives of the inverse trigonometric functions are as follows:

ddxarccot(x)=−11+x2\dfrac{d}{dx} \text{arccot}(x) = \dfrac {-1} {1 + x^2}​dx​​d​​arccot(x)=​1+x​2​​​​−1​​

ddxarcsec(x)=1x21−x−2\dfrac{d}{dx} \text{arcsec}(x) = \dfrac {1} {x^2\sqrt{1 - x^{-2}}}​dx​​d​​arcsec(x)=​x​2​​√​1−x​−2​​​​​​​1​​

ddxarccsc(x)=−1x21−x−2\dfrac{d}{dx} \text{arccsc}(x) = \dfrac {-1} {x^2\sqrt{1 - x^{-2}}}​dx​​d​​arccsc(x)=​x​2​​√​1−x​−2​​​​​​​−1​​

Note that some of these functions are not valid for a range of xxx which would end up making the function undefined.

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