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𝑛
𝑛− 1
𝑛
𝑛− 1
𝑛
𝑛− 1 𝑑 (
2
𝑢
𝑢
𝑑(ln 𝑢)
𝑢
𝑢
𝑎
𝑢 ln 𝑎
= cos (𝑢)
𝑑(cot 𝑢)
= −csc
2
= −sen (𝑢)
𝑑(sec 𝑢)
= sec(𝑢) ∙ tan ( 𝑢)
= sec
2
𝑑(csc 𝑢)
= −csc(𝑢) ∙ cot (𝑢)
2
𝑎𝑟𝑐 cot 𝑢
2
2
𝑎𝑟𝑐 sec 𝑢
2
2
𝑑(arc csc 𝑢)
2
− 1
𝑛
𝑛+ 1
= ln|𝑢| + 𝑐
𝑢
𝑢
- 𝑐 ∫ ln 𝑢 𝑑𝑢 = 𝑢 ln (𝑢) − 𝑢 + 𝑐
𝑢
𝑢
ln 𝑎
𝑛
ln 𝑢 𝑑𝑢 =
𝑛+ 1
2
[(𝑛 + 1 ) ln 𝑢 − 1 ] + 𝑐
∫ 𝑠𝑒𝑛 𝑢 𝑑𝑢 = − cos(𝑢) + 𝑐 ∫ cot 𝑢 𝑑𝑢 = ln|sen 𝑢| + 𝑐
∫ cos 𝑢 𝑑𝑢 = 𝑠𝑒𝑛 (𝑢) + 𝑐 ∫ sec 𝑢 𝑑𝑢 = ln|sec(𝑢) + tan(𝑢)| + 𝑐
∫ tan 𝑢 𝑑𝑢 = ln|sec 𝑢| + 𝑐 ∫ csc 𝑢 𝑑𝑢 = ln|csc
− cot
∫ sec 𝑢 ∙ tan 𝑢 𝑑𝑢 = sec(𝑢) + 𝑐 ∫ csc 𝑢 ∙ cot 𝑢 𝑑𝑢 = − csc(𝑢) + 𝑐
2
2
𝑢 𝑑𝑢 = − cot
2
2
𝑢 𝑑𝑢 = tan
2
𝑢 𝑑𝑢 = tan(𝑢) − 𝑢 + 𝑐 ∫ 𝑐𝑠𝑐
2
𝑢 𝑑𝑢 = − cot(𝑢) + 𝑐
3
2
( 𝑢
) cos(𝑢) + 𝑐 ∫ 𝑐𝑜𝑡
3
2
( 𝑢
− ln
sen 𝑢
3
2
sen(𝑢) + 𝑐 ∫ 𝑠𝑒𝑐
3
ln[sec(𝑢) + tan(𝑢)] + 𝑐
3
2
(𝑢) + ln|cos 𝑢| + 𝑐 ∫ 𝑐𝑠𝑐
3
ln|csc(𝑢) − cot(𝑢)| + 𝑐
𝑛
𝑛− 1
(𝑢) ∙ cos(𝑢) +
𝑛− 2
𝑛
𝑛− 1
𝑛− 2
𝑛
𝑛− 1
(𝑢) ∙ sen(𝑢) +
𝑛− 2
𝑛
𝑛− 2
(𝑢) ∙ tan(𝑢) +
𝑛− 2
𝑛
𝑛− 1
𝑛− 2
𝑛
𝑛− 2
(𝑢) ∙ cot(𝑢) +
𝑛− 2
2
− 𝑢
2
, 𝑎 > 0
2
2
− 1
2
2
2
2
2
− 1
2
− 𝑢
2
ln |
2
− 𝑢
2
2
√ 𝑎
2
2
2
2
)√ 𝑎
2
2
4
− 1
2
2
2
2
𝑢
2
2
2
− 𝑢
2
2
− 𝑢
2
− 𝑎 ln |
2
− 𝑢
2
2
− 𝑎
2
, 𝑎 > 0
2
± 𝑎
2
= ln |𝑢 + √𝑢
2
± 𝑎
2
| + 𝑐 ∫ √𝑢
2
− 𝑎
2
𝑑𝑢 =
2
− 𝑎
2
−
2
ln |𝑢 + √𝑢
2
− 𝑎
2
| + 𝑐
log 𝑎𝑏 = log 𝑎 + log 𝑏
log
= log 𝑎 − log 𝑏
log 𝑎
= 𝑛 log 𝑎
log 𝑎
log 𝑎
𝑛
𝑛
𝑠𝑒𝑛 𝜃 =
csc 𝜃
cos 𝜃 =
sec 𝜃
𝑡𝑎𝑛 𝜃 =
cot 𝜃
cot 𝜃 =
tan 𝜃
𝑐𝑠𝑐 𝜃 =
sen 𝜃
sec 𝜃 =
cos 𝜃
𝑠𝑒𝑛
𝜃 + 𝑐𝑜𝑠
𝜃 = 1
𝑠𝑒𝑛
𝜃 = 1 − 𝑐𝑜𝑠
𝜃 𝑐𝑜𝑠
𝜃 = 1 − 𝑠𝑒𝑛
𝜃
𝑠𝑒𝑐
𝜃 = 1 + 𝑡𝑎𝑛
𝜃 𝑡𝑎𝑛
𝜃 = 𝑠𝑒𝑐
𝜃 − 1
𝑐𝑠𝑐
𝜃 = 1 + 𝑐𝑜𝑡
𝜃 𝑐𝑜𝑡
𝜃 = 𝑐𝑠𝑐
𝜃 − 1
𝑡𝑎𝑛 𝜃 =
cos 𝜃
cot 𝜃 =
cos 𝜃
sen 𝜃
𝑠𝑒𝑛 2 𝜃 = 2 𝑠𝑒𝑛𝜃 cos 𝜃 𝑐𝑜𝑠 2 𝜃 = 𝑠𝑒𝑛
𝜃 − 𝑐𝑜𝑠
𝜃
𝑡𝑎𝑛 2 𝜃 =
2
𝑐𝑜𝑠 2 𝜃 = 2 𝑐𝑜𝑠
𝜃 − 1
𝑠𝑒𝑛
𝜃 =
cos 2 𝜃 𝑐𝑜𝑠 2 𝜃 = 1 − 2 𝑠𝑒𝑛
𝜃
𝑡𝑎𝑛
𝜃 =
1 −cos 2 𝜃
1 +cos 2 𝜃
𝑠𝑒𝑛𝜃 ∙ csc 𝜃 = 1
𝑆𝑒𝑛 𝐴
𝑎
=
𝑆𝑒𝑛 𝐵
𝑏
=
𝑆𝑒𝑛 𝐶
𝑐
𝑎
= 𝑏
− 2 𝑏𝑐 cos 𝐴
𝑏
= 𝑎
− 2 𝑏𝑐 cos 𝐵
𝑐
= 𝑎
− 2 𝑏𝑐 cos 𝐶
𝑥 =
−𝑏 ± √𝑏
− 4 𝑎𝑐
2 𝑎
