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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 $= (-1)/(a(m-1)sin^(m-1)ax*cos^(n-1)ax)+(m+n-2)/(m-1)int 1/(sin^(m-2)ax*cos^n ax) dx$

Table of Integrals - Forms Involving sinaxsin ax and cosaxcos ax