∫
x^3 sinax dx=( 3 x 2
a^2 −6
a^4)
sinax+( 6 x
a −x^3
a)
cosax+C∫
xnsinax dx=−a^1 xncosax+an 2 xn−^1 sinax−n(na− 2 1)∫
xn−^2 sinax dx∫ sinax
x dx=ax−a^3 x^3
3 ·3!+a^5 x^5
5 ·5!−a^7 x^7
7 ·7!+···+(−1)nx^2 n+1x^2 n+1
(2n+ 1)·(2n+ 1)!+···∫ sinax
x^2 dx=−1
asinax+a∫ cosax
x dx∫ sinax
x^3 dx=−a
2 xcosax−1
2 x^2 sinax−a^2
2∫ sinax
x dx∫ sinax
xn dx=−sinax
(n−1)xn−^1 +a
n− 1∫ cosax
xn−^1 dx∫ dx
sinax=1
aln|cscas−cotax|+C∫ x dx
sinax=1
a^2[
ax+a(^3) x 3
18 +
7 a^5 x^5
1800 +···+
2(2^2 n−^1 −1)Bna^2 n+1x^2 n+1
(2n+ 1)! +···
]
+C
whereBnis the nthBernoulli numberB 1 = 1/ 6 ,B 2 = 1/ 30 ,.. .Note scaling and shifting
393.
∫ dx
xsinax=−
1
ax+
ax
6 +
7 a^3 x^3
1080 +···+
2(2^2 n−^1 −1)Bna^2 n+1x^2 n+1
(2n−1)(2n)! +···+C
394.
∫
sin^2 ax dx=x 2 −sin 2 4 aax+C
395.
∫
xsin^2 ax dx=x
2
4 −
xsin2ax
4 a −
cos 2ax
8 a^2 +C
396.
∫
x^2 sin^2 ax dx= 61 a− 4 a^12 cos 2ax+ 241 a 3 (3− 6 a^2 x^2 ) sin 2ax+C
397.
∫
sin^3 ax dx=−cosaax+cos
(^2) ax
3 a +C
398.
∫
xsin^3 ax dx= 121 axcos 3ax− 361 a 2 sin 3ax− 43 axcosax+ 43 a 2 sinax+C
399.
∫
sin^4 ax dx=^38 x−sin 2 4 aax+sin 4 32 aax+C
400.
∫ dx
sin^2 ax=−
1
acotax+C
Appendix C