∫ sinax dx
cosax =1
aln|secax|+C∫ cosax dx
sinax =1
aln|sinax|+C∫ xsinax dx
cosax =1
a^2[
a^3 x^3
3 +a^5 x^5
5 +2 a^7 x^7
105 +···+22 n(2^2 n−1)Bna^2 n+1x^2 n+1
(2n+ 1)!]
+C∫ xcosax dx
sinax =1
a^2[
ax−a(^3) x 3
9 −
a^5 x^5
225 −···−
22 nBna^2 n+1x^2 n+1
(2n+ 1)! −···
]
+C
468.
∫ cosax dx
xsinax =−
1
ax−
ax
2 −
a^3 x^3
135 −···−
22 nBna^2 n−^1 x^2 n−^1
(2n−1)(2n)! −···+C
469.
∫ sinax
xcosaxdx=ax+
a^3 x^3
9 +
2 a^5 x^5
75 +···+
22 n(2^2 n−1)Bna^2 n−^1 x^2 n−^1
(2n−1)(2n)! +···+C
470.
∫ sin (^2) ax
cos^2 axdx=
1
atanax−x+C
471.
∫ cos (^2) ax
sin^2 axdx=−
1
acotax−x+C
472.
∫ xsin (^2) ax
cos^2 ax dx=
1
axtanax+
1
a^2 ln|cosax|−
1
2 x
(^2) +C
473.
∫ xcos (^2) ax
sin^2 ax
dx=−^1 axcotax+a^12 ln|sinax|−^12 x^2 +C
474.
∫ cosax
sinaxdx=
1
aln|sinax|+C
475.
∫ sin (^3) ax
cos^3 axdx=
1
2 atan
(^2) ax+^1
aln|cosax|+C
476.
∫ cos (^3) ax
sin^3 ax
dx=− 21 acot^2 ax−^1 aln|sinax|+C
477.
∫
sin(ax+b) sin(ax+β)dx=x 2 cos(b−β)− 41 asin(2ax+b+β) +C
478.
∫
sin(ax+b) cos(ax+β)dx=x 2 sin(b−β)− 41 acos(2ax+b+β) +C
479.
∫
cos(ax+b) cos(ax+β)dx=x 2 cos(b−β) + 41 asin(2ax+b+β) +C
480.
∫
sin^2 axcos^2 bx dx=
x
4 −
sin 2ax
8 a +
sin 2bx
8 b −
sin 2(a−b)x
16(a−b) −
sin2(a+b)x
16(a+b) +C, b^6 =a
x
8 −
sin 4ax
32 a +C, b=a
Appendix C