The integrals below involve `sin ax` and `cos ax`.
*****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) `int sin ax*cos ax dx = (sin^2 ax)/(2a)`
2) `int sin px*cosqx dx = -(cos(p-q)x)/(2(p-q))-(cos(p+q)x)/(2(p+q)`
3) `int sin^n ax*cos ax dx = (sin^(n+1)ax)/((n+1)a)`
**[If `n=-1`, see integral #1 in the table for forms involving `cot ax`]
4) `int cos^n ax*sin ax dx = -(cos^(n+1)ax)/((n+1)a)`
**[If `n=-1`, see integral #1 in the table for forms involving `tan ax`]
5) `int sin^2ax*cos^2ax dx = x/8-(sin4ax)/(32a)`
6) `int 1/(sin ax*cos ax) dx = 1/a ln tan ax`
7) `int 1/(sin^2 ax*cos^2 ax) dx = 1/a ln tan(pi/4+(ax)/2)-1/(a sin ax)`
8) `int 1/(sin ax*cos^2ax) dx = 1/a ln tan((ax)/2)+1/(a cos ax)`
9) `int 1/(sin^2ax*cos^2ax) dx = -(2 cot 2ax)/a`
10) `int (sin^2ax)/(cos ax) dx = -(sin ax)/a+1/a ln tan((ax)/2+pi/4)`
11) `int (cos^2 ax)/(sin ax) dx = (cos ax)/a+1/a ln tan((ax)/2)`
12) `int 1/(cos ax(1+-sin ax)) dx = {-+}1/(2a(1+-sin ax))+1/(2a)ln tan((ax)/2+pi/4)`
13) `int 1/(sin ax(1+-cos ax)) dx = +-1/(2a(1+-cos ax))+1/(2a)ln tan((ax)/2)`
14) `int 1/(sin ax+-cosax) dx = 1/(asqrt2)ln tan((ax)/2+-pi/8)`
15) `int (sin ax)/(sin ax+-cos ax) dx = x/2{-+}1/(2a)ln(sin ax+-cos ax)`
16) `int (cos ax)/(sin ax+-cos ax) dx = +-x/2+1/(2a)ln(sin ax+-cos ax_)`
17) `int (sin ax)/(p+q cos ax) dx = -1/(aq)ln(p+q cos ax)`
18) `int (cos ax)/(p+q sin ax) dx = 1/(aq)ln(p+q sin ax)`
19) `int (sin ax)/(p+q cos ax)^n dx = 1/(aq(n-1)(p+q cos ax)^(n-1))`
20) `int (cos ax)/(p+q sin ax)^n dx = (-1)/(aq(n-1)(p+q sin ax)^(n-1))`
21) `int 1/(p sin ax+q cos ax) dx = 1/(asqrt(p^2+q^2))ln tan((ax+tan^-1(q/p))/2)`
22) `int 1/(p sin ax+q cos ax+r) dx = 2/(asqrt(r^2-p^2-q^2))tan^-1((p+(r-q)tan ((ax)/2))/sqrt(r^2-p^2-q^2))`
OR `= 1/(asqrt(p^2+q^2-r^2))ln((p-sqrt(p^2+q^2-r^2)+(r-q)tan((ax)/2))/(p+sqrt(p^2+q^2-r^2)+(r-q)tan((ax)/2)))`
**[If `r=q`, see integral #23 in this table]
**[If `r^2=p^2+q^2`, see integral #24 in this table]
23) `int 1/(p sin ax+q(1+cos ax)) dx = 1/(ap)ln[q+p tan((ax)/2)]`
24) `int 1/(p sin ax+q cos ax+-sqrt(p^2+q^2)) dx = (-1)/(asqrt(p^2+q^2))tan[pi/4{-+}(ax+tan^-1(q/p))/2]`
25) `int 1/(p^2 sin^2 ax+q^2 cos^2 ax) dx = 1/(apq)tan^-1((p tan ax)/q)`
26) `int 1/(p^2 sin^2 ax-q^2 cos^2 ax) dx = 1/(2apq)ln((p tan ax-q)/(p tan ax+q))`
27) `int sin^m ax*cos^n ax dx = -(sin^(m-1) ax*cos^(n+1) ax)/(a(m+n))+(m-1)/(m+n)int sin^(m-2) ax*cos^n ax dx`
OR `= (sin^(m+1) ax*cos^(n-1) ax)/(a(m+n))+(n-1)/(m+n)int sin^m ax*cos^(n-2) ax dx`
28) `int (sin^m ax)/(cos^n ax) dx = (sin^(m-1)ax)/(a(n-1)cos^(n-1)ax)-(m-1)/(n-1)int(sin^(m-2)ax)/(cos^(n-2)ax) dx`
OR `= (sin^(m+1)ax)/(a(n-1)cos^(n-1)ax)-(m-n+2)/(n-1)int (sin^m ax)/(cos^(n-2)ax) dx`
OR `= (-sin^(m-1)ax)/(a(m-n)cos^(n-1)ax)+(m-1)/(m-n)int (sin^(m-2)ax)/(cos^nax) dx`
29) `int (cos^m ax)/(sin^nax) dx = (-cos^(m-1)ax)/(a(n-1)sin^(n-1)ax)-(m-1)/(n-1)int (cos^(m-2)ax)/(sin^(n-2)ax) dx`
OR `= (-cos^(m+1)ax)/(a(n-1)sin^(n-1)ax)-(m-n+2)/(n-1)int(cos^m ax)/(sin^(n-2)ax) dx`
OR `= (cos^(m-1) ax)/(a(m-n)sin^(n-1)ax)+(m-1)/(m-n)int(cos^(m-2)ax)/(sin^n ax) dx`
30) `int 1/(sin^m ax*cos^n ax) dx = 1/(a(n-1)sin^(m-1)ax*cos^(n-1)ax)+(m+n-2)/(n-1)int 1/(sin^m ax*cos^(n-2)ax) dx`
OR `= (-1)/(a(m-1)sin^(m-1)ax*cos^(n-1)ax)+(m+n-2)/(m-1)int 1/(sin^(m-2)ax*cos^n ax) dx`