Cao Yi

Indefinite Integrals of Trigonometric and Inverse Trigonometric Functions

Index

This document compiles indefinite integrals for powers of the six basic trigonometric functions (sin, cos, tan, cot, sec, csc) and their inverses (arcsin, arccos, arctan, arccot, arcsec, arccsc), in two forms:

Powers of Trig Functions ($trig^n(x)$)

  sin cos tan cot sec csc
1 $\int \sin(x) \mathrm{d}x$ $\int \cos(x) \mathrm{d}x$ $\int \tan(x) \mathrm{d}x$ $\int \cot(x) \mathrm{d}x$ $\int \sec(x) \mathrm{d}x$ $\int \csc(x) \mathrm{d}x$
2 $\int \sin^2(x) \mathrm{d}x$ $\int \cos^2(x) \mathrm{d}x$ $\int \tan^2(x) \mathrm{d}x$ $\int \cot^2(x) \mathrm{d}x$ $\int \sec^2(x) \mathrm{d}x$ $\int \csc^2(x) \mathrm{d}x$
3 $\int \sin^3(x) \mathrm{d}x$ $\int \cos^3(x) \mathrm{d}x$ $\int \tan^3(x) \mathrm{d}x$ $\int \cot^3(x) \mathrm{d}x$ $\int \sec^3(x) \mathrm{d}x$ $\int \csc^3(x) \mathrm{d}x$
4 $\int \sin^4(x) \mathrm{d}x$ $\int \cos^4(x) \mathrm{d}x$ $\int \tan^4(x) \mathrm{d}x$ $\int \cot^4(x) \mathrm{d}x$ $\int \sec^4(x) \mathrm{d}x$ $\int \csc^4(x) \mathrm{d}x$
5 $\int \sin^5(x) \mathrm{d}x$ $\int \cos^5(x) \mathrm{d}x$ $\int \tan^5(x) \mathrm{d}x$ $\int \cot^5(x) \mathrm{d}x$ $\int \sec^5(x) \mathrm{d}x$ $\int \csc^5(x) \mathrm{d}x$
6 $\int \sin^6(x) \mathrm{d}x$ $\int \cos^6(x) \mathrm{d}x$ $\int \tan^6(x) \mathrm{d}x$ $\int \cot^6(x) \mathrm{d}x$ $\int \sec^6(x) \mathrm{d}x$ $\int \csc^6(x) \mathrm{d}x$
$\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$
n $\int \sin^n(x) \mathrm{d}x$ $\int \cos^n(x) \mathrm{d}x$ $\int \tan^n(x) \mathrm{d}x$ $\int \cot^n(x) \mathrm{d}x$ $\int \sec^n(x) \mathrm{d}x$ $\int \csc^n(x) \mathrm{d}x$

(Table A)

References:

Trig of Polynomial Arguments ($trig(x^n)$)

  sin cos tan cot sec csc
1 $\int \sin(x) \mathrm{d}x$ $\int \cos(x) \mathrm{d}x$ $\int \tan(x) \mathrm{d}x$ $\int \cot(x) \mathrm{d}x$ $\int \sec(x) \mathrm{d}x$ $\int \csc(x) \mathrm{d}x$
2 $\int \sin(x^2) \mathrm{d}x$ $\int \cos(x^2) \mathrm{d}x$ $\int \tan(x^2) \mathrm{d}x$ $\int \cot(x^2) \mathrm{d}x$ $\int \sec(x^2) \mathrm{d}x$ $\int \csc(x^2) \mathrm{d}x$
3 $\int \sin(x^3) \mathrm{d}x$ $\int \cos(x^3) \mathrm{d}x$ $\int \tan(x^3) \mathrm{d}x$ $\int \cot(x^3) \mathrm{d}x$ $\int \sec(x^3) \mathrm{d}x$ $\int \csc(x^3) \mathrm{d}x$
4 $\int \sin(x^4) \mathrm{d}x$ $\int \cos(x^4) \mathrm{d}x$ $\int \tan(x^4) \mathrm{d}x$ $\int \cot(x^4) \mathrm{d}x$ $\int \sec(x^4) \mathrm{d}x$ $\int \csc(x^4) \mathrm{d}x$
5 $\int \sin(x^5) \mathrm{d}x$ $\int \cos(x^5) \mathrm{d}x$ $\int \tan(x^5) \mathrm{d}x$ $\int \cot(x^5) \mathrm{d}x$ $\int \sec(x^5) \mathrm{d}x$ $\int \csc(x^5) \mathrm{d}x$
6 $\int \sin(x^6) \mathrm{d}x$ $\int \cos(x^6) \mathrm{d}x$ $\int \tan(x^6) \mathrm{d}x$ $\int \cot(x^6) \mathrm{d}x$ $\int \sec(x^6) \mathrm{d}x$ $\int \csc(x^6) \mathrm{d}x$

(Table B)

Powers of Inverse Trig Functions ($arctrig^n(x)$)

  arcsin arccos arctan arccot arcsec arccsc
1 $\int \arcsin(x) \mathrm{d}x$ $\int \arccos(x) \mathrm{d}x$ $\int \arctan(x) \mathrm{d}x$ $\int \operatorname{arccot}(x) \mathrm{d}x$ $\int \operatorname{arcsec}(x) \mathrm{d}x$ $\int \operatorname{arccsc}(x) \mathrm{d}x$
2 $\int \arcsin^2(x) \mathrm{d}x$ $\int \arccos^2(x) \mathrm{d}x$ $\int \arctan^2(x) \mathrm{d}x$ $\int \operatorname{arccot}^2(x) \mathrm{d}x$ $\int \operatorname{arcsec}^2(x) \mathrm{d}x$ $\int \operatorname{arccsc}^2(x) \mathrm{d}x$
3 $\int \arcsin^3(x) \mathrm{d}x$ $\int \arccos^3(x) \mathrm{d}x$ $\int \arctan^3(x) \mathrm{d}x$ $\int \operatorname{arccot}^3(x) \mathrm{d}x$ $\int \operatorname{arcsec}^3(x) \mathrm{d}x$ $\int \operatorname{arccsc}^3(x) \mathrm{d}x$
4 $\int \arcsin^4(x) \mathrm{d}x$ $\int \arccos^4(x) \mathrm{d}x$ $\int \arctan^4(x) \mathrm{d}x$ $\int \operatorname{arccot}^4(x) \mathrm{d}x$ $\int \operatorname{arcsec}^4(x) \mathrm{d}x$ $\int \operatorname{arccsc}^4(x) \mathrm{d}x$
5 $\int \arcsin^5(x) \mathrm{d}x$ $\int \arccos^5(x) \mathrm{d}x$ $\int \arctan^5(x) \mathrm{d}x$ $\int \operatorname{arccot}^5(x) \mathrm{d}x$ $\int \operatorname{arcsec}^5(x) \mathrm{d}x$ $\int \operatorname{arccsc}^5(x) \mathrm{d}x$
6 $\int \arcsin^6(x) \mathrm{d}x$ $\int \arccos^6(x) \mathrm{d}x$ $\int \arctan^6(x) \mathrm{d}x$ $\int \operatorname{arccot}^6(x) \mathrm{d}x$ $\int \operatorname{arcsec}^6(x) \mathrm{d}x$ $\int \operatorname{arccsc}^6(x) \mathrm{d}x$

(Table C)

Inverse Trig of Polynomial Arguments ($arctrig(x^n)$)

  arcsin arccos arctan arccot arcsec arccsc
1 $\int \arcsin(x) \mathrm{d}x$ $\int \arccos(x) \mathrm{d}x$ $\int \arctan(x) \mathrm{d}x$ $\int \operatorname{arccot}(x) \mathrm{d}x$ $\int \operatorname{arcsec}(x) \mathrm{d}x$ $\int \operatorname{arccsc}(x) \mathrm{d}x$
2 $\int \arcsin(x^2) \mathrm{d}x$ $\int \arccos(x^2) \mathrm{d}x$ $\int \arctan(x^2) \mathrm{d}x$ $\int \operatorname{arccot}(x^2) \mathrm{d}x$ $\int \operatorname{arcsec}(x^2) \mathrm{d}x$ $\int \operatorname{arccsc}(x^2) \mathrm{d}x$
3 $\int \arcsin(x^3) \mathrm{d}x$ $\int \arccos(x^3) \mathrm{d}x$ $\int \arctan(x^3) \mathrm{d}x$ $\int \operatorname{arccot}(x^3) \mathrm{d}x$ $\int \operatorname{arcsec}(x^3) \mathrm{d}x$ $\int \operatorname{arccsc}(x^3) \mathrm{d}x$
4 $\int \arcsin(x^4) \mathrm{d}x$ $\int \arccos(x^4) \mathrm{d}x$ $\int \arctan(x^4) \mathrm{d}x$ $\int \operatorname{arccot}(x^4) \mathrm{d}x$ $\int \operatorname{arcsec}(x^4) \mathrm{d}x$ $\int \operatorname{arccsc}(x^4) \mathrm{d}x$
5 $\int \arcsin(x^5) \mathrm{d}x$ $\int \arccos(x^5) \mathrm{d}x$ $\int \arctan(x^5) \mathrm{d}x$ $\int \operatorname{arccot}(x^5) \mathrm{d}x$ $\int \operatorname{arcsec}(x^5) \mathrm{d}x$ $\int \operatorname{arccsc}(x^5) \mathrm{d}x$
6 $\int \arcsin(x^6) \mathrm{d}x$ $\int \arccos(x^6) \mathrm{d}x$ $\int \arctan(x^6) \mathrm{d}x$ $\int \operatorname{arccot}(x^6) \mathrm{d}x$ $\int \operatorname{arcsec}(x^6) \mathrm{d}x$ $\int \operatorname{arccsc}(x^6) \mathrm{d}x$

(Table D)

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