BASIC DIFFERENTIATION
1. (๐ฅ ๐ ) = ๐ โ ๐ฅ ๐−1
2. (๐๐ฅ ๐ ) = ๐ โ ๐๐ฅ ๐−1
๐๐ข
๐๐ฃ
3. [๐ข ± ๐ฃ] = ๐๐ฅ ± ๐๐ฅ
4. (๐ข๐ฃ) = ๐ข๐๐ฃ + ๐ฃ๐๐ข
๐ข
5. ( )
๐ฃ
๐
6. (๐ฃ )
Log and fx properties
1. log ๐ ๐ข = ln ๐ข
2. log ๐ 1 = 0 (ln 1 = 0)
3. log ๐ ๐ = 1 (ln ๐ = 1)
4. ln ๐ฅ๐ฆ = ln ๐ฅ + ln ๐ฆ
๐ฅ
5. ln ๐ฆ = ln ๐ฅ − ln ๐ฆ
๐ฃ๐๐ข−๐ข๐๐ฃ
๐ฃ2
−๐
โ ๐๐ฃ
๐ฃ2
๐๐ข
2√๐ข
1
๐
1
๐) =
(๐ข
๐๐ฅ
๐
๐−1
=
=
7. √๐ข =
๐
8. √๐ข =
9. (๐ข๐ ) = ๐ โ ๐ข
a is
constant
Inverse trigo integ
1. ∫
โ๐ข
โ ๐๐ข
Exponential and logarithmic
๐๐ข
๐ข
1.log ๐ ๐ข = log ๐ ๐ โ
2. ๐๐ข = ๐๐ข ln ๐ โ ๐๐ข
๐๐ข
3. ln ๐ข =
๐ข
4. ๐ ๐ข = ๐ ๐ข ๐๐ข
๐๐ข
๐ ๐ฅ −๐ −๐ฅ
2
๐ ๐ฅ +๐ −๐ฅ
= 2
1. sinh ๐ฅ =
Trigonometric identities
1. ๐ ๐๐2 ∅ + ๐๐๐ 2 ∅ = 1
2. ๐ ๐๐ 2 ∅ − ๐ก๐๐2 ∅ = 1
3. ๐๐ ๐ 2 ∅ − ๐๐๐ก 2 ∅ = 1
4. sin 2∅ = 2 sin ∅ cos ∅
9. ∫
Trigonometric
1. sin ๐ข = cos ๐ข ๐๐ข
2. cos ๐ข = − sin ๐ข ๐๐ข
3. tan ๐ข = ๐ ๐๐ 2 ๐ข ๐๐ข
4. csc ๐ข = − csc ๐ข cot ๐ข ๐๐ข
5. sec ๐ข = sec ๐ข tan ๐ข ๐๐ข
6. cot ๐ข = −๐๐ ๐ 2 ๐ข ๐๐ข
6. cos 2∅ = 2 ๐๐๐ 2 ∅ − 1
= 1 − 2 ๐ ๐๐2 ∅
Inverse trigo
1. sin−1 ๐ข =
5. sinh ๐ฅ + ๐ฆ = sinh ๐ฅ cosh ๐ฆ +
cosh ๐ฅ sinh ๐ฆ
2. cos−1 ๐ข = −
1
๐๐ข
√1−๐ข2
4. csc −1 ๐ข = −
5. sec
−1
๐ข=
6. cot −1 ๐ข =
1
๐ข√๐ข2 −1
1
๐๐ข
7. cosh ๐ฅ − sinh ๐ฅ = ๐ −๐ฅ
8. cosh ๐ฅ + sinh ๐ฅ = ๐ ๐ฅ
9. ๐๐๐กโ2 ๐ฅ − 1 = ๐๐ ๐โ2 ๐ฅ
๐๐ข
Hyperbholic
1. sinh ๐ข = cosh ๐ข ๐๐ข
2. cosh ๐ข = sinh ๐ข ๐๐ข
3. tanh ๐ข = ๐ ๐๐โ2 ๐ข ๐๐ข
4. csch ๐ข = − csch ๐ข coth ๐ข ๐๐ข
5. sech ๐ข = − sech ๐ข tanh ๐ข ๐๐ข
6. coth ๐ข = −๐๐ ๐โ2 ๐ข ๐๐ข
cos
sin
cot
tan
csc
sec
c
Inverse hyperbolic
1. sinh−1 ๐ข =
2. cosh
−1
๐ข=
3. tanh−1 ๐ข =
4. csch−1 ๐ข =
5. sech−1 ๐ข =
−1
6. coth
๐ข
1
√1+๐ข2
1
๐๐ข
๐๐ข
√๐ข2 −1
1
๐๐ข
1−๐ข2
−1
|๐ข|√1−๐ข2
−1
๐๐ข
๐๐ข
๐ข√1−๐ข2
1
= 1−๐ข2 ๐๐ข
Hyperbolic identities
1. sinh −๐ข = − sinh ๐ข
2. cosh −๐ข = − cosh ๐ข
3. ๐๐๐ โ2 ๐ข − ๐ ๐๐โ2 ๐ข = 1
4. 1 − ๐ก๐๐โ2 ๐ข = ๐ ๐๐โ2 ๐ข
6. cosh ๐ฅ + ๐ฆ = cosh ๐ฅ cosh ๐ฆ +
sinh ๐ฅ sinh ๐ฆ
๐ข√๐ข2 −1
1
− 1+๐ข2 ๐๐ข
1
๐๐ข
๐ข2 −๐2
๐ข
=
๐ข√๐ข2 −1
๐๐ข
10. ∫
๐๐ข
3. tan−1 ๐ข = 1+๐ข2 ๐๐ข
๐ข
1
๐ข−๐
ln |
|
2๐
๐ข+๐
1
5. ∫ √๐ข2 + ๐2 ๐๐ข = ๐ข√๐ข2 + ๐2 +
2
1 2
2
๐ ln |๐ข + √๐ข + ๐2 |
2
1
6. ∫ √๐2 − ๐ข2 ๐๐ข = ๐ข√๐2 − ๐ข2 +
2
1 2
−1 ๐ข
๐
sin
2
๐
1
2
7.∫ √๐ข − ๐2 ๐๐ข = 2 ๐ข√๐ข2 − ๐2 −
1
ln |๐ข + √๐ข2 − ๐2
2
๐๐ข
8. ∫
= tan−1 ๐ข
1+๐ข2
๐๐ข
−1
4. ∫
2 tan ∅
√1−๐ข2
1
1
3.∫
= ๐ sec −1 ๐
๐ข√๐ข2 −๐2
5. tan 2∅ = 1−๐ก๐๐2 ∅
1
= sin−1 ๐
๐๐ข
6. ln ๐ฅ ๐ฆ = ๐ฆ ln ๐ฅ
2. cosh ๐ฅ
โ ๐๐ข
๐ข
√๐ 2 −๐ข2
2.∫ ๐2 +๐ข2 = ๐ tan−1 ๐
Hyperbolic properties
1
−1
๐
๐๐ข
√1−๐ข2
= sec
๐ข
= sin−1 ๐ข
Hyperbolic integ
1. ∫ sinh ๐ข ๐๐ข = cosh ๐ข
2. ∫ cosh ๐ข ๐๐ข = sinh ๐ข
3. ∫ ๐ ๐๐โ2 ๐ข ๐๐ข = tanh ๐ข
4. ∫ ๐๐ ๐โ2 ๐ข ๐๐ข = − coth ๐ข
5. ∫ sech ๐ข tanh ๐ข ๐๐ข = − sech ๐ข
6. ∫ csch ๐ข coth ๐ข ๐๐ข = − csch ๐ข
7. ∫ tanh ๐ข ๐๐ข = ln | cosh ๐ข|
8. ∫ coth ๐ข ๐๐ข = ln | sinh ๐ข|
9. ∫ sech ๐ข ๐๐ข = tan−1 (sinh ๐ข)
๐ข
10. ∫ csch ๐ข ๐๐ข = ln | tanh |
2
By parts
Basic integration
1. ∫ ๐๐ข = ๐ข
2. ∫ ๐๐ข ๐๐ข = ๐ ∫ ๐ข ๐๐ข
3. ∫ ๐ข๐ =
∫ ๐ข ๐๐ฃ = ๐ข๐ฃ − ∫ ๐ฃ๐๐ข
l-log
i-inv. Trigo
a-algebra
t-trigo
e-exponent, e
๐ข๐+1
๐+1
Exponential and log integ
๐๐ข
1. ∫ ๐ข = ln |๐ข|
2. ∫ ๐ ๐ข ๐๐ข = ๐ ๐ข
3. ∫ ๐๐ข ๐๐ข =
๐๐ข
ln |๐ข|
Trigo integ
1. ∫ sin ๐ข ๐๐ข = − cos ๐ข
2. ∫ cos ๐ข ๐๐ข = sin ๐ข
3. ∫ tan ๐ข ๐๐ข = ln | sec ๐ข|
4. ∫ cot ๐ข ๐๐ข = ln | sin ๐ข|
5. ∫ sec ๐ข ๐๐ข = ln | sec ๐ข + tan ๐ข|
6. ∫ csc ๐ข ๐๐ข = ln | csc ๐ข − cot ๐ข |
7. ∫ sec ๐ข tan ๐ข ๐๐ข = sec ๐ข
8. ∫ csc ๐ข cot ๐ข ๐๐ข = − csc ๐ข
1
1
9. ∫ ๐ ๐๐2 ๐ข ๐๐ข = 2 ๐ข − 4 sin 2๐ข
1
1
10. ∫ ๐๐๐ 2 ๐ข ๐๐ข = 2 ๐ข + 4 sin 2๐ข
11. ∫ ๐ก๐๐2 ๐ข ๐๐ข = tan ๐ข − ๐ข
12. ∫ ๐๐๐ก 2 ๐ข ๐๐ข = − cot ๐ข − ๐ข
13. ∫ ๐ ๐๐ 2 ๐ข ๐๐ข = tan ๐ข
14. ∫ ๐๐ ๐ 2 ๐ข ๐๐ข = − cot ๐ข
0