∫ sinax dx
sinax±cosax=x
2 ∓ln|sinax±cosax|+C∫ cosax dx
sinax±cosax=±x
2 +1
2 aln|sinaxx±cosax|+C∫ sinax dx
α+βsinax=1
aβln|α+βsinax|+C∫ cosax dx
α+βsinax=1
aβln|α+βsinax|+C∫ sinaxcosax dx
α^2 cos^2 ax+β^2 ax=1
2 a(β^2 −α^2 )ln|α(^2) cos (^2) ax+β (^2) sin (^2) ax|+C, β 6 =α
500.
∫ dx
α^2 sin^2 ax+β^2 cos^2 ax=
1
aαβtan
− 1
(α
βtanax
)
+C
501.
∫ dx
α^2 sin^2 ax−β^2 cos^2 ax=
1
2 aαβln
∣∣
∣∣αtanax−β
αtanax+β
∣∣
∣∣+C
502.
∫ sinnax
cos(n+2axdx=
tann+1ax
(n+ 1)a +C
503.
∫ cosnax
sin(n+2)ax
dx=−cot
(n+1)ax
(n+ 1)a +C
504.
∫ dx
α+βsincosaxax
=α 2 αx+β 2 +a(α 2 β+β (^2) )ln|βsinax+αcosax|+C
505.
∫ dx
α+βcossinaxax
=α 2 αx+β 2 −a(α 2 β+β (^2) )ln|αsinax+βcosax|+C
506.
∫ cosnax
sinnaxdx=−
cot(n−1)ax
(n−1)a −
∫
cot(n−2)ax dx
507.
∫ sinnax
cosnaxdx=
tann−^1 ax
(n−1)a −
∫ sinn− (^2) ax
cosn−^2 axdx
508.
∫ sinax
cos(n+1)axdx=
1
nasec
nax+C
509.
∫ αsinx+βcosx
γsinx+δcosxdx=
[(αγ+βδ)x+ (βγ−αγ) ln|γsinx+δcosx|]
γ^2 +δ^2 +C
510.
∫ α+βsinx
a+bcosxdx=
√^2 α
a^2 −b^2
tan−^1
√
a−b
a+btan
x
2 −
β
bln|a+bcosx|+C, a > b
√^2 α
b^2 −a^2
tanh−^1
√
b−a
b+atan
x
2 −
β
bln|a+bcosx|+C, a < b
Appendix C