BASIC DIFFERENTIATION
1. (𝑥 𝑛 ) = 𝑛 ∙ 𝑥 𝑛−1
2. (𝑎𝑥 𝑛 ) = 𝑎 ∙ 𝑛𝑥 𝑛−1
𝑑𝑢
𝑑𝑣
3. [𝑢 ± 𝑣] = 𝑑𝑥 ± 𝑑𝑥
4. (𝑢𝑣) = 𝑢𝑑𝑣 + 𝑣𝑑𝑢
𝑢
5. ( )
𝑣
𝑎
6. (𝑣 )
Log and fx properties
1. log 𝑒 𝑢 = ln 𝑢
2. log 𝑎 1 = 0 (ln 1 = 0)
3. log 𝑎 𝑎 = 1 (ln 𝑒 = 1)
4. ln 𝑥𝑦 = ln 𝑥 + ln 𝑦
𝑥
5. ln 𝑦 = ln 𝑥 − ln 𝑦
𝑣𝑑𝑢−𝑢𝑑𝑣
𝑣2
−𝑎
∙ 𝑑𝑣
𝑣2
𝑑𝑢
2√𝑢
1
𝑑
1
𝑛) =
(𝑢
𝑑𝑥
𝑛
𝑛−1
=
=
7. √𝑢 =
𝑛
8. √𝑢 =
9. (𝑢𝑛 ) = 𝑛 ∙ 𝑢
a is
constant
Inverse trigo integ
1. ∫
∙𝑢
∙ 𝑑𝑢
Exponential and logarithmic
𝑑𝑢
𝑢
1.log 𝑎 𝑢 = log 𝑎 𝑒 ∙
2. 𝑎𝑢 = 𝑎𝑢 ln 𝑎 ∙ 𝑑𝑢
𝑑𝑢
3. ln 𝑢 =
𝑢
4. 𝑒 𝑢 = 𝑒 𝑢 𝑑𝑢
𝑑𝑢
𝑒 𝑥 −𝑒 −𝑥
2
𝑒 𝑥 +𝑒 −𝑥
= 2
1. sinh 𝑥 =
Trigonometric identities
1. 𝑠𝑖𝑛2 ∅ + 𝑐𝑜𝑠 2 ∅ = 1
2. 𝑠𝑒𝑐 2 ∅ − 𝑡𝑎𝑛2 ∅ = 1
3. 𝑐𝑠𝑐 2 ∅ − 𝑐𝑜𝑡 2 ∅ = 1
4. sin 2∅ = 2 sin ∅ cos ∅
9. ∫
Trigonometric
1. sin 𝑢 = cos 𝑢 𝑑𝑢
2. cos 𝑢 = − sin 𝑢 𝑑𝑢
3. tan 𝑢 = 𝑠𝑒𝑐 2 𝑢 𝑑𝑢
4. csc 𝑢 = − csc 𝑢 cot 𝑢 𝑑𝑢
5. sec 𝑢 = sec 𝑢 tan 𝑢 𝑑𝑢
6. cot 𝑢 = −𝑐𝑠𝑐 2 𝑢 𝑑𝑢
6. cos 2∅ = 2 𝑐𝑜𝑠 2 ∅ − 1
= 1 − 2 𝑠𝑖𝑛2 ∅
Inverse trigo
1. sin−1 𝑢 =
5. sinh 𝑥 + 𝑦 = sinh 𝑥 cosh 𝑦 +
cosh 𝑥 sinh 𝑦
2. cos−1 𝑢 = −
1
𝑑𝑢
√1−𝑢2
4. csc −1 𝑢 = −
5. sec
−1
𝑢=
6. cot −1 𝑢 =
1
𝑢√𝑢2 −1
1
𝑑𝑢
7. cosh 𝑥 − sinh 𝑥 = 𝑒 −𝑥
8. cosh 𝑥 + sinh 𝑥 = 𝑒 𝑥
9. 𝑐𝑜𝑡ℎ2 𝑥 − 1 = 𝑐𝑠𝑐ℎ2 𝑥
𝑑𝑢
Hyperbholic
1. sinh 𝑢 = cosh 𝑢 𝑑𝑢
2. cosh 𝑢 = sinh 𝑢 𝑑𝑢
3. tanh 𝑢 = 𝑠𝑒𝑐ℎ2 𝑢 𝑑𝑢
4. csch 𝑢 = − csch 𝑢 coth 𝑢 𝑑𝑢
5. sech 𝑢 = − sech 𝑢 tanh 𝑢 𝑑𝑢
6. coth 𝑢 = −𝑐𝑠𝑐ℎ2 𝑢 𝑑𝑢
cos
sin
cot
tan
csc
sec
c
Inverse hyperbolic
1. sinh−1 𝑢 =
2. cosh
−1
𝑢=
3. tanh−1 𝑢 =
4. csch−1 𝑢 =
5. sech−1 𝑢 =
−1
6. coth
𝑢
1
√1+𝑢2
1
𝑑𝑢
𝑑𝑢
√𝑢2 −1
1
𝑑𝑢
1−𝑢2
−1
|𝑢|√1−𝑢2
−1
𝑑𝑢
𝑑𝑢
𝑢√1−𝑢2
1
= 1−𝑢2 𝑑𝑢
Hyperbolic identities
1. sinh −𝑢 = − sinh 𝑢
2. cosh −𝑢 = − cosh 𝑢
3. 𝑐𝑜𝑠ℎ2 𝑢 − 𝑠𝑖𝑛ℎ2 𝑢 = 1
4. 1 − 𝑡𝑎𝑛ℎ2 𝑢 = 𝑠𝑒𝑐ℎ2 𝑢
6. cosh 𝑥 + 𝑦 = cosh 𝑥 cosh 𝑦 +
sinh 𝑥 sinh 𝑦
𝑢√𝑢2 −1
1
− 1+𝑢2 𝑑𝑢
1
𝑑𝑢
𝑢2 −𝑎2
𝑢
=
𝑢√𝑢2 −1
𝑑𝑢
10. ∫
𝑑𝑢
3. tan−1 𝑢 = 1+𝑢2 𝑑𝑢
𝑢
1
𝑢−𝑎
ln |
|
2𝑎
𝑢+𝑎
1
5. ∫ √𝑢2 + 𝑎2 𝑑𝑢 = 𝑢√𝑢2 + 𝑎2 +
2
1 2
2
𝑎 ln |𝑢 + √𝑢 + 𝑎2 |
2
1
6. ∫ √𝑎2 − 𝑢2 𝑑𝑢 = 𝑢√𝑎2 − 𝑢2 +
2
1 2
−1 𝑢
𝑎
sin
2
𝑎
1
2
7.∫ √𝑢 − 𝑎2 𝑑𝑢 = 2 𝑢√𝑢2 − 𝑎2 −
1
ln |𝑢 + √𝑢2 − 𝑎2
2
𝑑𝑢
8. ∫
= tan−1 𝑢
1+𝑢2
𝑑𝑢
−1
4. ∫
2 tan ∅
√1−𝑢2
1
1
3.∫
= 𝑎 sec −1 𝑎
𝑢√𝑢2 −𝑎2
5. tan 2∅ = 1−𝑡𝑎𝑛2 ∅
1
= sin−1 𝑎
𝑑𝑢
6. ln 𝑥 𝑦 = 𝑦 ln 𝑥
2. cosh 𝑥
∙ 𝑑𝑢
𝑢
√𝑎 2 −𝑢2
2.∫ 𝑎2 +𝑢2 = 𝑎 tan−1 𝑎
Hyperbolic properties
1
−1
𝑛
𝑑𝑢
√1−𝑢2
= sec
𝑢
= sin−1 𝑢
Hyperbolic integ
1. ∫ sinh 𝑢 𝑑𝑢 = cosh 𝑢
2. ∫ cosh 𝑢 𝑑𝑢 = sinh 𝑢
3. ∫ 𝑠𝑒𝑐ℎ2 𝑢 𝑑𝑢 = tanh 𝑢
4. ∫ 𝑐𝑠𝑐ℎ2 𝑢 𝑑𝑢 = − coth 𝑢
5. ∫ sech 𝑢 tanh 𝑢 𝑑𝑢 = − sech 𝑢
6. ∫ csch 𝑢 coth 𝑢 𝑑𝑢 = − csch 𝑢
7. ∫ tanh 𝑢 𝑑𝑢 = ln | cosh 𝑢|
8. ∫ coth 𝑢 𝑑𝑢 = ln | sinh 𝑢|
9. ∫ sech 𝑢 𝑑𝑢 = tan−1 (sinh 𝑢)
𝑢
10. ∫ csch 𝑢 𝑑𝑢 = ln | tanh |
2
By parts
Basic integration
1. ∫ 𝑑𝑢 = 𝑢
2. ∫ 𝑎𝑢 𝑑𝑢 = 𝑎 ∫ 𝑢 𝑑𝑢
3. ∫ 𝑢𝑛 =
∫ 𝑢 𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣𝑑𝑢
l-log
i-inv. Trigo
a-algebra
t-trigo
e-exponent, e
𝑢𝑛+1
𝑛+1
Exponential and log integ
𝑑𝑢
1. ∫ 𝑢 = ln |𝑢|
2. ∫ 𝑒 𝑢 𝑑𝑢 = 𝑒 𝑢
3. ∫ 𝑎𝑢 𝑑𝑢 =
𝑎𝑢
ln |𝑢|
Trigo integ
1. ∫ sin 𝑢 𝑑𝑢 = − cos 𝑢
2. ∫ cos 𝑢 𝑑𝑢 = sin 𝑢
3. ∫ tan 𝑢 𝑑𝑢 = ln | sec 𝑢|
4. ∫ cot 𝑢 𝑑𝑢 = ln | sin 𝑢|
5. ∫ sec 𝑢 𝑑𝑢 = ln | sec 𝑢 + tan 𝑢|
6. ∫ csc 𝑢 𝑑𝑢 = ln | csc 𝑢 − cot 𝑢 |
7. ∫ sec 𝑢 tan 𝑢 𝑑𝑢 = sec 𝑢
8. ∫ csc 𝑢 cot 𝑢 𝑑𝑢 = − csc 𝑢
1
1
9. ∫ 𝑠𝑖𝑛2 𝑢 𝑑𝑢 = 2 𝑢 − 4 sin 2𝑢
1
1
10. ∫ 𝑐𝑜𝑠 2 𝑢 𝑑𝑢 = 2 𝑢 + 4 sin 2𝑢
11. ∫ 𝑡𝑎𝑛2 𝑢 𝑑𝑢 = tan 𝑢 − 𝑢
12. ∫ 𝑐𝑜𝑡 2 𝑢 𝑑𝑢 = − cot 𝑢 − 𝑢
13. ∫ 𝑠𝑒𝑐 2 𝑢 𝑑𝑢 = tan 𝑢
14. ∫ 𝑐𝑠𝑐 2 𝑢 𝑑𝑢 = − cot 𝑢