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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) $int x/(sin^n ax) dx = (-xcos ax)/(a(n-1)sin^(n-1) ax)-1/(a^2(n-1)(n-2)sin^(n-2)ax)+(n-2)/(n-1)int x/(sin^(n-2) ax) dx$

Table of Integrals - Forms Involving sinaxsin ax