∫ dx
sinaxcosax=1
aln|tanax|+C∫ dx
sin^2 axcosax=1
aln|tan(π
4 +ax
2)
|−asin^1 ax+C∫ dx
sinaxcos^2 ax=1
aln|tanax
2 |+1
acosax+C∫ dx
sin^2 axcos^2 ax=−2 cos 2a ax+C∫ sin (^2) ax
cosaxdx=−
sinax
a +
1
aln|tan
(ax
2 +
π
4
)
|+C
486.
∫ cos (^2) ax
sinax dx=
cosax
a +
1
aln|tan
ax
2 |+C
487.
∫ dx
cosax(1 + sinax)=
1
2 a(1 + sinax)
[
−1 + (1 + sinax) ln
∣∣
∣∣cos
ax 2 + sinax 2
cosax 2 −sinax 2
∣∣
∣∣
]
+C
488.
∫ dx
sinax(1 + cosax)=
1
4 asec
2 ax
2 +
1
2 aln|tan
ax
2 |+C
489.
∫ dx
sinax(α+βsinax)=
1
aαln|tan
ax
2 |−
β
α
∫ dx
α+βsinax
490.
∫ dx
cosax(α+βsinax=
1
α^2 −β^2
[
α
aln|tan
(π
4 +
ax
2
)
|−βαln
∣∣
∣∣α+βsinax
cosax
∣∣
∣∣
]
+C, β 6 =α
491.
∫ dx
sinax(α+βcosax)=
1
α^2 −β^2
[
α
aln|tan
ax
2 |+
β
aln|
α+βcosax
sinax |
]
+C, β 6 =α
492.
∫ dx
cosax(α+βcosax)=
1
aαln|tan
(π
4 +
ax
2
)
|−βα
∫ dx
α+βcosax
493.
∫ dx
α+βcosax+γsinax=
2
a
√
−R
tan−^1
(γ+ (α−β) tanax
√^2
−R
)
+C, α
(^2) > β (^2) +γ 2
R=β^2 +γ^2 −α^2
1
a
√
R
ln
∣∣
∣∣
∣
γ−
√
R+ (α−β) tanax 2
γ+
√
R+ (α−β) tanax 2
∣∣
∣∣
∣+C, α
(^2) < β (^2) +γ 2
1
aβln
∣∣
∣β+γtanax 2
∣∣
∣+C, α=β
1
aβln
∣∣
∣∣ cos
ax 2 + sinax 2
(β+γ) cosax 2 + (γ−β) sinax 2
∣∣
∣∣+C, α=γ
1
aγln|1 + tan
ax
2 |+C, α=β=γ
494.
∫ dx
sinax±cosax=
√^1
2 a
ln|tan
(ax
2 ±
π
8
)
|+C
Appendix C