Table of Integrals - Forms Involving `sin ax`

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`