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Table of Integrals - Forms Involving sinax and cosax

The integrals below involve sinax and cosax.

*****Special Note: In integrals #12, #15, and #24 you will see this symbol {-+}. This is to be read as "minus or plus".

Treat this as you would treat the ± symbol with the order reversed. For example, a±b=c±d means that a+b=c+d OR that a-b=c-d.

In contrast, a±b=c{-+}d means that a+b=c-d OR that a-b=c+d.

1)  sinaxcosax dx=sin2ax2a

2)  sinpxcosqx dx=-cos(p-q)x2(p-q)-cos(p+q)x2(p+q)

3)  sinnaxcosax dx=sinn+1ax(n+1)a

                    **[If n=-1, see integral #1 in the table for forms involving cotax]

4)  cosnaxsinax dx=-cosn+1ax(n+1)a

                    **[If n=-1, see integral #1 in the table for forms involving tanax]

5)  sin2axcos2ax dx=x8-sin4ax32a

6)  1sinaxcosax dx=1alntanax

7)  1sin2axcos2ax dx=1alntan(π4+ax2)-1asinax

8)  1sinaxcos2ax dx=1alntan(ax2)+1acosax

9)  1sin2axcos2ax dx=-2cot2axa

10)  sin2axcosax dx=-sinaxa+1alntan(ax2+π4)

11)  cos2axsinax dx=cosaxa+1alntan(ax2)

12)  1cosax(1±sinax) dx={-+}12a(1±sinax)+12alntan(ax2+π4)

13)  1sinax(1±cosax) dx=±12a(1±cosax)+12alntan(ax2)

14)  1sinax±cosax dx=1a2lntan(ax2±π8)

15)  sinaxsinax±cosax dx=x2{-+}12aln(sinax±cosax)

16)  cosaxsinax±cosax dx=±x2+12aln(sinax±cosax)

17)  sinaxp+qcosax dx=-1aqln(p+qcosax)

18)  cosaxp+qsinax dx=1aqln(p+qsinax)

19)  sinax(p+qcosax)n dx=1aq(n-1)(p+qcosax)n-1

20)  cosax(p+qsinax)n dx=-1aq(n-1)(p+qsinax)n-1

21)  1psinax+qcosax dx=1ap2+q2lntan(ax+tan-1(qp)2)

22)  1psinax+qcosax+r dx=2ar2-p2-q2tan-1(p+(r-q)tan(ax2)r2-p2-q2)

                                           OR =1ap2+q2-r2ln(p-p2+q2-r2+(r-q)tan(ax2)p+p2+q2-r2+(r-q)tan(ax2))

                    **[If r=q, see integral #23 in this table]

                    **[If r2=p2+q2, see integral #24 in this table]

23)  1psinax+q(1+cosax) dx=1apln[q+ptan(ax2)]

24)  1psinax+qcosax±p2+q2 dx=-1ap2+q2tan[π4{-+}ax+tan-1(qp)2]

25)  1p2sin2ax+q2cos2ax dx=1apqtan-1(ptanaxq)

26)  1p2sin2ax-q2cos2ax dx=12apqln(ptanax-qptanax+q)

27)  sinmaxcosnax dx=-sinm-1axcosn+1axa(m+n)+m-1m+n sinm-2axcosnax dx

                                 OR =sinm+1axcosn-1axa(m+n)+n-1m+n sinmaxcosn-2 ax dx

28)  sinmaxcosnax dx=sinm-1axa(n-1)cosn-1ax-m-1n-1sinm-2axcosn-2ax dx

                     OR =sinm+1axa(n-1)cosn-1ax-m-n+2n-1sinmaxcosn-2ax dx

                     OR =-sinm-1axa(m-n)cosn-1ax+m-1m-nsinm-2axcosnax dx

29)  cosmaxsinnax dx=-cosm-1axa(n-1)sinn-1ax-m-1n-1cosm-2axsinn-2ax dx

                     OR =-cosm+1axa(n-1)sinn-1ax-m-n+2n-1cosmaxsinn-2ax dx

                     OR =cosm-1axa(m-n)sinn-1ax+m-1m-ncosm-2axsinnax dx

30)  1sinmaxcosnax dx=1a(n-1)sinm-1axcosn-1ax+m+n-2n-11sinmaxcosn-2ax dx

                                  OR =-1a(m-1)sinm-1axcosn-1ax+m+n-2m-11sinm-2axcosnax dx