∫ x dx
sin^2 ax=−x
acotax+1
a^2 ln|sinax|+C∫ dx
sin^3 ax=−cosax
2 asin^2 ax+1
2 aln|tanax
2 |+C∫ dx
sinnax=−cosax
(n−1)asinn−^1 ax+nn−−^21∫ dx
sinn−^2 ax∫ dx
1 −sinax=1
atan(π
4 −ax
2)
+C∫ dx
a−sinax=2
a√
a^2 − 1tan−^1[atan(ax/2)− 1
√
a^2 − 1]
+C, a > 1∫ x dx
1 −sinax=x
atan(π
4 −ax
2)
+a^22 ln|sin(π
4 −ax
2)
|+C∫ dx
1 + sinax=−1
atan(π
4 −ax
2)
+C∫ dx
a+ sinax=2
a√
a^2 − 1tan−^1[1 +atan(ax/2)
√
a^2 − 1]
+C, a > 1∫ x dx
1 + sinax=x
atan(π
4 −ax
2)
+a^22 ln|sin(π
4 −ax
2)
|+C∫ dx
1 + sin^2 x=√^1
2tan−^1 (√
2 tanx) +C∫ dx
1 −sin^2 x= tanx+C∫ dx
(1−sinax)^2 =1
2 atan(π
4 −ax
2)
+ 61 atan^3(π
4 −ax
2)
+C∫ dx
(1 + sinax)^2 =−1
2 atan(π
4 −ax
2)
− 61 atan^3(π
4 −ax
2)
+C∫ dx
α+βsinax=
2
a√
α^2 −β^2tan−^1(
αtanax 2 +β)
+C, α^2 > β^21
a√
β^2 −α^2ln∣∣
∣∣
∣αtanax 2 +β−√
β^2 −α^2
αtanax 2 +β+√
β^2 −α^2∣∣
∣∣
∣+C, α(^2) < β 2
1
aαtan
(ax
2 ±
π
4
)
+C, β=±α
415.
∫ dx
α^2 +β^2 sin^2 ax
=^1
aα
√
β^2 +α^2
tan−^1
(√
β^2 +α^2
α tanax
)
+C
Appendix C