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Table of Integrals - Forms Involving cosax

The integrals below involve cosax.

1)  cosax dx=sinaxa

2)  xcosax dx=cosaxa2+xsinaxa

3)  x2cosax dx=2xa2cosax+(x2a-2a3)sinax

4)  x3cosax dx=(3x2a2-6a4)cosax+(x3a-6xa3)sinax

5)  cosaxx dx=lnx-(ax)222!+(ax)444!-(ax)666!+...

6)  cosaxx2 dx=-cosaxx-a sinaxx dx

                    **[See integral #5 in the previous table; forms involving sinax]

7)  1cosax dx=1aln(secax+tanax)=1alntan(π4+ax2)

8)  xcosax dx=1a2{(ax)22+(ax)48+5(ax)6144+...+En(ax)2n+2(2n+2)(2n)!+...}

9)  cos2ax dx=x2+sin2ax4a

10)  xcos2ax dx=x24+xsin2ax4a+cos2ax8a2

11)  cos3ax dx=sinaxa-sin3ax3a

12)  cos4ax dx=3x8+sin2ax4a+sin4ax32a

13)  1cos2ax dx=tanaxa

14)  1cos3ax dx=sinax2acos2ax+12alntan(π4+ax2)

15)  cosaxcospx dx=sin(a-p)x2(a-p)+sin(a+p)x2(a+p)

                    **[If a=±p, see integral #9 in this table]

16)  11-cosax dx=-1acot(ax2)

17)  x1-cosax dx=-xacot(ax2)+2a2lnsin(ax2)

18)  11+cosax dx=1atan(ax2)

19)  x1+cosax dx=xatan(ax2)+2a2lncos(ax2)

20)  1(1-cosax)2 dx=-12acot(ax2)-16acot3(ax2)

21)  1(1+cosax)2 dx=12atan(ax2)+16atan3(ax2)

22)  1p+qcosax dx=2ap2-q2tan-1p-qp+qtan(12ax)

                           OR =1aq2-p2ln(tan(12ax)+q+pq-ptan(12ax)-q+pq-p)

                    **[If p=±q, see integrals #16 and #18 in this table]

23)  1(p+qcosax)2 dx=qsinaxa(q2-p2)(p+qcosax)-pq2-p2 1p+qcosax dx

                    **[If p=±q, see integrals #20 and #21 in this table]

24)  1p2+q2cos2ax dx=1app2+q2tan-1(ptanaxp2+q2)

25)  1p2-q2cos2ax dx=1app2-q2tan-1(ptanaxp2-q2)

                               OR =12apq2-p2ln(ptanax-q2-p2ptanax+q2-p2)

26)  xmcosax dx=xmsinaxa+mxm-1a2cosax-m(m-1)a2 xm-2cosax dx

27)  cosaxxn dx=-cosax(n-1)xn-1-an-1 sinaxxn-1 dx

                    **[See integral #27 in the previous table; forms involving sinax]

28)  cosnax dx=sinaxcosn-1 axan+n-1n cosn-2ax dx

29)  1cosnax dx=sinaxa(n-1)cosn-1ax+n-2n-1 1cosn-2ax dx

30)  xcosnax dx=xsinaxa(n-1)cosn-1ax-1a2(n-1)(n-2)cosn-2ax+n-2n-1 xcosn-2ax dx