Table of Integrals - Forms Involving `sin ax` and `cos ax`

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`