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) ∫ sinax⋅cosax dx=sin2ax2a
2) ∫ sinpx⋅cosqx dx=-cos(p-q)x2(p-q)-cos(p+q)x2(p+q)
3) ∫ sinnax⋅cosax dx=sinn+1ax(n+1)a
**[If n=-1, see integral #1 in the table for forms involving cotax]
4) ∫ cosnax⋅sinax dx=-cosn+1ax(n+1)a
**[If n=-1, see integral #1 in the table for forms involving tanax]
5) ∫ sin2ax⋅cos2ax dx=x8-sin4ax32a
6) ∫ 1sinax⋅cosax dx=1alntanax
7) ∫ 1sin2ax⋅cos2ax dx=1alntan(π4+ax2)-1asinax
8) ∫ 1sinax⋅cos2ax dx=1alntan(ax2)+1acosax
9) ∫ 1sin2ax⋅cos2ax 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=1a√2lntan(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=1a√p2+q2lntan(ax+tan-1(qp)2)
22) ∫ 1psinax+qcosax+r dx=2a√r2-p2-q2tan-1(p+(r-q)tan(ax2)√r2-p2-q2)
OR =1a√p2+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=-1a√p2+q2tan[π4{-+}ax+tan-1(qp)2]
25) ∫ 1p2sin2ax+q2cos2ax dx=1apqtan-1(ptanaxq)
26) ∫ 1p2sin2ax-q2cos2ax dx=12apqln(ptanax-qptanax+q)
27) ∫ sinmax⋅cosnax dx=-sinm-1ax⋅cosn+1axa(m+n)+m-1m+n∫ sinm-2ax⋅cosnax dx
OR =sinm+1ax⋅cosn-1axa(m+n)+n-1m+n∫ sinmax⋅cosn-2 ax dx
28) ∫ sinmaxcosnax dx=sinm-1axa(n-1)cosn-1ax-m-1n-1∫sinm-2axcosn-2ax dx
OR =sinm+1axa(n-1)cosn-1ax-m-n+2n-1∫sinmaxcosn-2ax dx
OR =-sinm-1axa(m-n)cosn-1ax+m-1m-n∫sinm-2axcosnax dx
29) ∫ cosmaxsinnax dx=-cosm-1axa(n-1)sinn-1ax-m-1n-1∫cosm-2axsinn-2ax dx
OR =-cosm+1axa(n-1)sinn-1ax-m-n+2n-1∫cosmaxsinn-2ax dx
OR =cosm-1axa(m-n)sinn-1ax+m-1m-n∫cosm-2axsinnax dx
30) ∫ 1sinmax⋅cosnax dx=1a(n-1)sinm-1ax⋅cosn-1ax+m+n-2n-1∫1sinmax⋅cosn-2ax dx
OR =-1a(m-1)sinm-1ax⋅cosn-1ax+m+n-2m-1∫1sinm-2ax⋅cosnax dx