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4 Elementary FunctionsTrigonometric Functions

§4.26 Integrals

Contents

§4.26(i) Introduction

Throughout this section the variables are assumed to be real. The results in §§4.26(ii) and 4.26(iv) can be extended to the complex plane by using continuous branches and avoiding singularities.

§4.26(ii) Indefinite Integrals

4.26.1 sinxdx =-cosx,
4.26.2 cosxdx =sinx.
4.26.3 tanxdx =-ln(cosx),
-12π<x<12π.
4.26.4 cscxdx =ln(tan12x),
0<x<π.
4.26.5 secxdx=gd-1(x),
-12π<x<12π.

For the right-hand side see (4.23.41) and (4.23.42).

4.26.6 cotxdx=ln(sinx),
0<x<π.
4.26.7 eaxsin(bx)dx=eaxa2+b2(asin(bx)-bcos(bx)),
4.26.8 eaxcos(bx)dx=eaxa2+b2(acos(bx)+bsin(bx)).

§4.26(iii) Definite Integrals

Throughout this subsection m and n are integers.

Orthogonality Properties

4.26.9 0πsin(mt)sin(nt)dt=0,
mn,
4.26.10 0πcos(mt)cos(nt)dt=0,
mn,
4.26.11 0πsin2(nt)dt=0πcos2(nt)dt=12π,
n0.
4.26.12 0sin(mt)tdt={12π,m>0,0,m=0,-12π,m<0.
4.26.13 0sin(t2)dt=0cos(t2)dt=12π2.

§4.26(iv) Inverse Trigonometric Functions

4.26.14 arcsinxdx=xarcsinx+(1-x2)1/2,
-1<x<1,
4.26.15 arccosxdx=xarccosx-(1-x2)1/2,
-1<x<1.
4.26.16 arctanxdx=xarctanx-12ln(1+x2),
-<x<,
4.26.17 arccscxdx=xarccscx+ln(x+(x2-1)1/2),
1<x<,
4.26.18 arcsecxdx=xarcsecx-ln(x+(x2-1)1/2),
1<x<,
4.26.19 arccotxdx=xarccotx+12ln(1+x2),
0<x<.
4.26.20 xarcsinxdx=(x22-14)arcsinx+x4(1-x2)1/2,
-1<x<1,
4.26.21 xarccosxdx=(x22-14)arccosx-x4(1-x2)1/2,
-1<x<1.

§4.26(v) Compendia

Extensive compendia of indefinite and definite integrals of trigonometric and inverse trigonometric functions include Apelblat (1983, pp. 48–109), Bierens de Haan (1939), Gradshteyn and Ryzhik (2000, Chapters 2–4), Gröbner and Hofreiter (1949, pp. 116–139), Gröbner and Hofreiter (1950, pp. 94–160), and Prudnikov et al. (1986a, §§1.5, 1.7, 2.5, 2.7).