The integrals below involve `sin ax`
1) `int sin ax dx = -(cos ax)/a`
2) `int x sinax dx = (sin ax)/a^2-(xcos ax)/a`
3) `int x^2 sin ax dx = (2x)/a^2 sin ax +(2/a^3-x^2/a) cos ax`
4) `int x^3 sin ax dx = ((3x^2)/a^2-6/a^4)sin ax+((6x)/a^3-x^3/a)cos ax`
5) `int (sin ax)/x dx = ax-(ax)^3/(3*3!)+(ax)^5/(5*5!)-...`
6) `int (sin ax)/x^2 dx = -(sin ax)/x+a int (cos ax)/x dx`
**[See integral #5 in the next table; forms involving `cos ax`]
7) `int 1/(sin ax) dx = 1/a ln(csc ax-cot ax) = 1/a ln tan((ax)/2)`
8) `int x/(sin ax) dx = 1/a^2{ax+(ax)^3/18+(7(ax)^5)/1800+...+(2(2^(2n-1)-1)B_n(ax)^(2n+1))/((2n+1)!)+...}`
9) `int sin^2 ax dx = x/2-(sin2ax)/(4a)`
10) `int x*sin^2 ax dx = x^2/4-(xsin2ax)/(4a)-(cos 2ax)/(8a^2)`
11) `int sin^3 ax dx = -(cosax)/a+(cos^3 ax)/(3a)`
12) `int sin^4 ax dx = (3x)/8-(sin 2ax)/(4a)+(sin 4ax)/(32a)`
13) `int 1/(sin^2 ax) dx = -1/a cot ax`
14) `int 1/(sin^3 ax) dx = -(cos ax)/(2a*sin^2 ax)+1/(2a) ln tan((ax)/2)`
15) `int sin px *sin qx dx = (sin(p-q)x)/(2(p-q))-(sin(p+q)x)/(2(p+q))`
**[If `p=+-q`, see integral #9 in this table]
16) `int 1/(1-sin ax) dx = 1/a tan(pi/4+(ax)/2)`
17) `int x/(1-sin ax) dx = x/atan(pi/4+(ax)/2)+2/a^2 ln sin(pi/4-(ax)/2)`
18) `int 1/(1+sin ax) dx = -1/atan(pi/4-(ax)/2)`
19) `int x/(1+sin ax) dx = -x/atan(pi/4-(ax)/2)+2/(a^2) ln sin(pi/4+(ax)/2)`
20) `int 1/(1-sin ax)^2 dx = 1/(2a)tan(pi/4+(ax)/2)+1/(6a)tan^3(pi/4+(ax)/2)`
21) `int 1/(1+sin ax)^2 dx = -1/(2a)tan(pi/4-(ax)/2)-1/(6a)tan^3(pi/4-(ax)/2)`
22) `int 1/(p+q sin ax) dx = 2/(asqrt(p^2-q^2))tan^-1((p tan(1/2ax)+q)/sqrt(p^2-q^2))`
OR `= 1/(asqrt(q^2-p^2) ) ln((p tan(1/2ax)+q-sqrt(q^2-p^2))/(p tan (1/2ax)+q+sqrt(q^2-p^2)))`
**[If `p=+-q`, see integrals #16 and #18 in this table]
23) `int 1/(p+q sin ax)^2 dx = (q cos ax)/(a(p^2-q^2)(p+q sin ax))+p/(p^2-q^2) int 1/(p+q sin ax) dx`
**[If `p=+-q`, see integrals #20 and #21 in this table]
24) `int 1/(p^2+q^2 sin^2 ax) dx = 1/(apsqrt(p^2+q^2))tan^-1((sqrt(p^2+q^2) tan ax)/p)`
25) `int 1/(p^2-q^2 sin^2 ax) dx = 1/(apsqrt(p^2-q^2))tan^-1((sqrt(p^2-q^2) tan ax)/p)`
OR `= 1/(2apsqrt(q^2-p^2)) ln ((sqrt(q^2-p^2) tan ax+p)/(sqrt(q^2-p^2) tan ax-p))`
26) `int x^m sin ax dx = -(x^m cos ax)/a+(mx^(m-1)sin ax)/a^2-(m(m-1))/a^2 int x^(m-2) sin ax dx`
27) `int (sin ax)/x^n dx-(sin ax)/((n-1)x^(n-1))+a/(n-1)int (cos ax)/x^(n-1) dx`
**[See integral #27 in the next table; forms involving `cos ax`]
28) `int sin^n ax dx = -(sin^(n-1) ax cos ax)/(an)+(n-1)/nint sin^(n-2)ax dx`
29) `int 1/(sin^nax) dx = (-cos ax)/(a(n-1)sin^(n-1)ax)+(n-2)/(n-1)int 1/(sin^(n-2)ax) dx`
30) `int x/(sin^n ax) dx = (-xcos ax)/(a(n-1)sin^(n-1) ax)-1/(a^2(n-1)(n-2)sin^(n-2)ax)+(n-2)/(n-1)int x/(sin^(n-2) ax) dx`