∫ dx
(1−cosax)^2 =−1
2 acotax
2 −1
6 acot3 ax
2 +C∫ dx
(1 + cosax)^2 =1
2 atanax
2 +1
6 atan2 ax
2 +C∫ dx
α+βcosax=
2
a√
α^2 −β^2tan−^1(√
α−β
α+βtanax
2)
+C, α^2 > β^21
a√
β^2 −α^2ln∣∣
∣∣√
β+α+√
√ β−αtanax^2
β+α−√β−αtanax 2∣∣
∣∣+C, α^2 < β^2∫ dxα+cosβax=xα−βα∫ dx
β+αcosax∫ dx
(α+βcosax)^2 =αsinax
a(β^2 −α^2 )(α+βcosax)−α
β^2 −α^2∫ dx
α+βcosax, α^6 =β∫ dx
α^2 +β^2 cos^2 ax=1
aα√
α^2 +β^2tan−^1(
√αtanax
α^2 +β^2)
+C∫ dx
α^2 −β^2 cos^2 ax=
1
aα√
α^2 −β^2tan−^1(
√αtanax
α^2 −β^2)
+C, α^2 > β^21
2 aα√
β^2 −α^2ln∣∣
∣∣
∣αtanax−√
β^2 −α^2
αtanax+√
β^2 −α^2∣∣
∣∣
∣+C, α(^2) < β 2
456.
∫ dx
cosnax=
sec(n−2)axtanax
(n−1)a +
n− 2
n− 1
∫
secn−^2 ax dx+C
Integrals containing both sine and cosine functions
457.
∫
sinaxcosax dx= 21 asin^2 ax+C
458.
∫ dx
sinaxcosax=−
1
aln|cotax|+C
459.
∫
sinaxcosbx dx=−cos(2(aa−−bb))x−cos(2(aa++bb))x+C, a 6 =b
460.
∫
sinaxsinbx dx=sin(2(aa−−bb))x−sin(2(aa++bb))x+C
461.
∫
cosaxcosbx dx=sin(2(aa−−bb))x+sin(2(aa++bb))x+C
462.
∫
sinnaxcosax dx=sin
n+1ax
(n+ 1)a +C
463.
∫
cosnaxsinax dx=−cos
n+1ax
(n+ 1)a +C
Appendix C