Antidifferentiation

1.1 Antidifferentiation or Indefinite Integrals

 Antidifferentiation is the process of getting the antiderivatives.

 If F is an antiderivative of f, we write ∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝐶.
 The symbol ∫ , called the integral sign, denotes the operation of antidifferentiation.
 The function f is called the integrand.

Standard Integration Formulas

5 Basic Integration Formulas

1) ∫ du = u + C Topics
2) ∫(u + v)dx = ∫ udx + ∫ vdx A. Indefinite Integrals
3) ∫ audx = a ∫ udx  Powers
xn+1
4) ∫ x n dx = + C; n ≠ −1  Logartithms Functions
n+1
5) ∫
du
dx = ln|u| + C  Exponential Functions
u
 Trigonometric Functions
Logarithmic & Exponential Formulas  Transformation by
au Trigonometric Functions
6) ∫ au du = +C
ln⁡|a|  Inverse Trigonometric Functions
7) ∫ eu ⁡du = eu + C  Integration by parts
B. Integration by Substitution
Trigonometric Formulas
C. Integration of rational Fractions
8) ∫ cos u⁡du⁡ = sin u + C D. Wallis’ Formulas
9) ∫ sin u⁡du = − cos u + C E. Plane Areas Improper Integrals
10) ∫ sec 2 ⁡u⁡du = tan u + C
11) ∫ csc 2 u⁡du = ⁡ − cot u + C Practice Problems with Answer
12) ∫ sec u tan u⁡du = sec u + C
13) ∫ csc u cot u⁡du = − csc u + C
14) ∫ tanudu = −In cos u + C = In sec u + C
15) ∫ cot u⁡du = ln⁡ cos u + C = In sec u + C Hyperbolic Functions Formulas
16) ∫ sec u⁡du = ln⁡(sec u + tan u) + C
=- In (sec u – tan u) +C 1) ∫ cosh u⁡du = sinh u + C
17) ∫ csc⁡u⁡du = In⁡(csc u − cot u) + C 2) ∫ sinh u⁡du = cosh u + C
=- In (csc u +cot u) + C 3) ∫ sech2 u⁡du = tanhu + C
4) ∫ csc 2 h⁡u⁡du = −cothu + C
Additional Formulas
5) ∫ sech(u) tanh(u) = − sech(u) + C
du u 6) ∫ csch(u) coth(u) = − csch(u) + C
18) ∫ = Arcsin a + C
√a2 −u 2
du 1 u
19) ∫ a2⁡ +u2 = a ⁡Arctan a + C
du 1 u
20) ∫ = ⁡Arcsec a + C
u⁡√a2 −u2 a
du 1 𝑢−𝑎
21) ∫ u2⁡ −a2 = 2a ⁡In 𝑢+𝑎 + 𝐶⁡
𝑑𝑢 1 𝑢−𝑎
22) ∫ 𝑎2 −𝑢2 = 2a ⁡In 𝑢+𝑎 + 𝐶

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 𝑑𝑢
23) ∫ = In(u + √u2 ∓ a2 ) + 𝐶⁡
√u2 ±𝑎2
24) ∫ u⁡dv = uv − ∫ v⁡du

Indefinite Integral
Integrate the following:

Powers, Exponential, Logarithmic, Trigonometric Functions

3𝑥
1. ∫ 𝑑𝑥
4𝑥+5
(𝑥 2 +1⁡)2
2. ∫ 𝑑𝑥
𝑥+1
3
3. ∫ √16𝑛2 + 8𝑛 + 1 𝑑𝑛
4
4. ∫ 𝑥 2 (𝑥 −3 + 2) 𝑑𝑥
𝑥
5. ∫ √4 + √𝑥 𝑑𝑥
𝑥+2
6. ∫ 𝑑𝑥
𝑥+1
1+𝑒 𝑥
7. ∫ 𝑑𝑥
1−𝑒 𝑥
8. ∫ sin2 𝑥 𝑐𝑜𝑠𝑥 𝑑𝑥
9. ∫ 𝑡𝑎𝑛𝑥 𝑑𝑥
1
10. ∫ 𝑑𝑥
1−𝑐𝑜𝑠𝑥
11. ∫ (𝑡𝑎𝑛2𝑥 + 𝑠𝑒𝑐2𝑥)2 𝑑𝑥
1−𝑐𝑜𝑠2𝑦
12. ∫ 𝑑𝑦
1+𝑐𝑜𝑠2𝑦

Transformation by Trigonometric Functions

 Power of Sines and Cosines
 Power of Tangent and Secant
 Power of Cotangent and Cosecant
For Power of Sines and Cosines
Case I. When m is a positive odd integer and n is any number, we may write
sin𝑚 𝑣 cos 𝑛 𝑣 = (sin𝑚−1 𝑣 cosn 𝑣⁡)𝑠𝑖𝑛𝑣
sin2 𝑥 = 1 −cos2 𝑥
Case II. When m is any number and n is a positive odd we may write
sin𝑚 𝑣 cos𝑛 𝑣 = (sin𝑚 𝑣 cos n−1 𝑣⁡)𝑐𝑜𝑠𝑣
cos 2 𝑥 = 1 − sin2 𝑥
Case III. When m and n are both even integers (either both positive or one positive and one zero)
𝑛 𝑛
sin𝑚 𝑣 cos 𝑛 𝑣 = (sin2 𝑣) 2 ⁡𝑜𝑟⁡(cos 2 𝑣) 2
1 − 𝑐𝑜𝑠2𝑣 1 + 𝑐𝑜𝑠2𝑣
sin2 𝑣 = , cos 2𝑣 =
2 2

13. ∫ sin3 4𝑥𝑐𝑜𝑠 2 4𝑥⁡𝑑𝑥

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 sin3 𝑥⁡𝑑𝑥
14. ∫
cos6 𝑥
15. ∫ sin 𝑦𝑐𝑜𝑠 3 𝑦⁡𝑑𝑦
3

16. ∫ cos 3 𝑥⁡𝑑𝑥

17. ∫ sin4 4𝑥⁡𝑑𝑥

For Power of Tangents and Secants

Case I. When m is any number and n is a positive even integer greater than 2, we may write
tan𝑚 𝑣 sec 𝑛 𝑣 = (tan𝑚 𝑣 sec n−2 𝑣⁡) sec 2 𝑣
sec 2 𝑥 = 1 −tan2 𝑥
Case II. When m is any number and n is a positive even integer greater than 2, we may write
tan𝑚 𝑣 sec 𝑛 𝑣 = (tan𝑚−1 𝑣 sec n−1 𝑣⁡) 𝑠𝑒𝑐 𝑣 tan 𝑣
tan2 𝑥 = sec 2 𝑥 − 1
Case III. When m is a positive odd (or even) integer and n is zero, we may write
tan𝑚 𝑣 sec 𝑛 𝑣 = tan𝑚−2 𝑣𝑡𝑎𝑛2 𝑣
tan2 𝑥 = sec 2 𝑥 − 1

18. ∫ tan3 𝑥𝑠𝑒𝑐 4 𝑥⁡𝑑𝑥

19. ∫ tan3 𝑥𝑠𝑒𝑐 5 𝑥⁡𝑑𝑥
20. ∫ tan3 𝑥𝑑𝑥

Trigonometric Substitutions/ Inverse Trigonometric Substitution / Additional

Formulas
3 Theorem needed to consider:
 If the integrand contains a2 − 𝑢2 , use the substitution u=asinϴ
 If the integrand contains u2 + 𝑎2 , use the substitution u=atanϴ
 If the integrand contains u2 − 𝑎2 , use the substitution u=asecϴ

𝑑𝑥
21. ∫
𝑥√9+4𝑥 2
𝑥 2 𝑑𝑥
22. ∫
√2𝑥−𝑥 2
𝑑𝑡
23. ∫
√5−16𝑡 2
𝑦 2 +1⁡𝑑𝑦
24. ∫
𝑦 2 +4⁡

Integration by parts

∫ 𝑢𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣𝑑𝑢

L-Logarithmic
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I-Inverse
A-Algebraic
T-Trigonometric
E-Exponential

25.⁡∫ 𝑡 2 𝑐𝑜𝑠𝑡⁡𝑑𝑡
26. ∫ 𝑦lny dy
27. ∫ 𝑙𝑛𝑥⁡𝑑𝑥
28. ∫ 𝑥 2 𝑒 −𝑥 𝑑𝑥
29. ∫ 𝐴𝑟𝑐𝑡𝑎𝑛𝑥⁡𝑑𝑥
30. ∫ 𝑥𝑒 𝑥 𝑑𝑥

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Integration of Rational Fractions

 Distinct Linear Fractions
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 Repeated Linear Fractions
 Distinct Quadratic Fractions
 Repeated Quadratic Fractions

Distinct Linear Fractions

(2𝑥+11)𝑑𝑥
31. ∫
𝑥 2 +𝑥−6
3𝑥 2 +8𝑥−12⁡
32. ∫ 𝑑𝑥
𝑥 3 +7𝑥 2 +12𝑥⁡
3𝑡 2 𝑑𝑡
32.⁡∫
𝑡 4 +5𝑡 2 +4

Repeated Linear Fractions

𝑑𝑥
33. ∫
𝑥(𝑥+2)2

Distinct Quadratic Factors

𝑥⁡𝑑𝑥
34.⁡∫
𝑥 2 +6𝑥+13
sec2 ϴ⁡𝑑ϴ
35.∫
tan3 ϴ+4tanϴ

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Wallis’ Formula
𝝅
(𝒎−𝟏)(𝒎−𝟑)….(𝟐𝒐𝒓𝟏)∗(𝒏−𝟏)(𝒏−𝟑)….(𝟐𝒐𝒓𝟏)
∫𝟎 𝒔𝒊𝒏𝒎 𝒙𝒄𝒐𝒔𝒏 𝒙𝒅𝒙 =
𝟐
(𝒎+𝒏)(𝒎+𝒏−𝟐)….(𝟐𝒐𝒓⁡𝟏)
(𝜶)

Where:
m and n are nonnegative integers
𝝅
𝛼 = if both m and n are even
𝟐
𝛼 = 1 if both one and both are odd.

𝝅
36. ∫0 sin4 ϴ𝑑ϴ
2
𝝅
37. ∫0 sin4 ϴ𝑐𝑜𝑠 3 ϴ𝑑ϴ
2

𝝅
38.⁡∫0 sin2 ϴ𝑐𝑜𝑠 6 ϴ𝑑ϴ
2

𝝅
39.⁡∫0 cos4 3ϴ⁡𝑑ϴ
6

Plane Areas Improper Integrals

Steps:
 Name of the curve
 Plotting
 Solve for the intersection
 Get the area
2 methods in getting the area:
 Horizontal Rectangle Strips
 Vertical Rectangle Strips

Find the area of the region bounded by the curves

40. y=2x, xy=6, x=6, x-axis

41.⁡y = √𝑥 2 + 5, 𝑥 − 𝑎𝑥𝑖𝑠, 𝑥 = 2⁡(𝑓𝑖𝑟𝑠𝑡⁡𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡)
42. x 2 = 4𝑦 and y 2 = −4𝑥

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