INTEGRAL CALCULUS
PART 1
ENGR. PETER BENJAMIN B. OBIANO
INTEGRAL CALCULUS 1
• Integration is an important concept in mathematics and together
with its inverse; differentiation, is one of the two main operations in
calculus. Integration can be considered as the reverse process of
differentiation.
• It is the process of finding a function given its derivative.
INTEGRAL CALCULUS 1
• ANTIDERIVATIVES
INTEGRAL CALCULUS I
• INDEFINITE INTEGRAL
INTEGRAL CALCULUS I
• BASIC RULES
INTEGRALS
/
PROPERTIES
OF
INDEFINITE
INTEGRAL CALCULUS I
• BASIC RULES
INTEGRALS
/
PROPERTIES
OF
INDEFINITE
INTEGRAL CALCULUS I
• BASIC RULES
INTEGRALS
/
PROPERTIES
OF
INDEFINITE
INTEGRAL CALCULUS I
• BASIC RULES
INTEGRALS
/
PROPERTIES
OF
INDEFINITE
INTEGRAL CALCULUS I
• BASIC RULES
INTEGRALS
/
PROPERTIES
OF
INDEFINITE
INTEGRAL CALCULUS I
• BASIC RULES
INTEGRALS
/
PROPERTIES
OF
INDEFINITE
INTEGRAL CALCULUS I
• BASIC RULES
INTEGRALS
/
PROPERTIES
OF
INDEFINITE
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
1. Evaluate: ‫ ׬‬6𝑥 5 − 18𝑥 2 + 7 𝑑𝑥
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
2. Evaluate: න 5 𝑧 − 4 𝑧 2 − 8𝑧
1
3 dz
INTEGRAL CALCULUS I
INTEGRAL OF NATURAL LOGARITHM
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
3. Evaluate: න
𝑑𝑥
𝑥𝑙𝑛 𝑥 3
INTEGRAL CALCULUS I
INTEGRAL OF EXPONENTIAL FUNCTIONS
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
4. Evaluate:
න𝑒
5𝑥+1
𝑑𝑥
INTEGRAL CALCULUS I
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
5. Evaluate:
sin x
න
dx
2
cos x
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
6. Evaluate:
2
න tan 𝑥 dx
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
7. Evaluate:
න 𝑥𝑠𝑖𝑛 3𝑥 2 dx
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
8. Evaluate:
cosh x
න
dx
2 + 3 sinh x
INTEGRAL CALCULUS I
INTEGRAL USING INVERSE
TRIGONOMETRIC
FUNCTIONS
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
9. Evaluate:
න
𝑑𝑥
4 − 𝑥2
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
𝑑𝑣
න
2 + 9𝑣 2
10. Evaluate:
INTEGRAL CALCULUS I
INTEGRATION BY PARTS
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
11. Evaluate:
𝑥
න 𝑥𝑒 𝑑𝑥
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
12. Evaluate:
𝑥
න 𝑒 cos 𝑥 𝑑𝑥
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
2
𝑥
ln
𝑥
+ 3 𝑑𝑥
13. Evaluate:
න
𝑥2 + 3
ln x 2 + 3
A.
4
2
ln x − 3
B.
4
2
+C
2
ln x 2 − 5
C.
2
2
+C
ln x + 5
D.
2
2
+C
2
+C
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
𝑠4 − 3 𝑠
14. Evaluate:න
𝑑𝑠
6 𝑠
1 9 1 5
A.
s2 + s6 + C
27
5
1 9 1 5
C.
s2 − s6 + C
27
5
1 7 1 5
B.
s2 − s6 + C
27
5
1 7 1 5
D.
s2 + s6 + C
27
5
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
cos3 𝑥
15. Evaluate:න
𝑑𝑥
1 − sin 𝑥
1
A. 𝑐𝑜𝑠 x + cos 2 x + C
2
1 2
C. sin x − sin x + C
4
1 2
B. sin x + sin x + C
2
1
D. cos x − cos 2 x + C
4
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
16. Evaluate:
8
න
1
2+𝑡
3
𝑡2
𝑑𝑡
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
𝑡
17. Evaluate: න 6𝑥 𝑥 − 1 𝑑𝑥
0
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
18. Evaluate
t
‫׬‬0 t
+ sin x dx
A. t^2 - cos t + 1
B. t^2/2 + t sin t
C. t^2 - sin t
D. t^2/2 + cos t
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
19. Evaluate:
π
2
න sin2 x cos 4 x dx
0
π
A.
32
π
B. −
32
32
C.
π
32
D. −
π
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
20. Evaluate:
𝜋 sin 𝑥
න න
0
0
4 + 𝑐𝑜𝑠𝑥 𝑑𝑦𝑑𝑥
INTEGRAL CALCULUS I
• PRACTICE PROBLEMS
21. Evaluate:
2 𝑥2 𝑦
න න න 𝑥𝑦𝑧 𝑑𝑧𝑑𝑦𝑑𝑥
0 −1 1
INTEGRAL CALCULUS I
IMPROPER INTEGRALS
𝑏
• Given a definite integral ‫𝑥𝑑 𝑥 𝑓 𝑎׬‬. An improper integral is a definite integral that
has a) either or both the limits a, b is/are infinite OR b) an integrand f that has an
infinite or essential discontinuity at in the interval [a,b].
•
• An improper integral is convergent if the defining limit exists (the value of the
defining limit is a finite number).That limit is the value of the improper integral.
• An improper integral is divergent if the defining limit does not exist (for example if
the value of the limiting integral is infinity).
INTEGRAL CALCULUS I
IMPROPER INTEGRALS
Integrand with Infinite Limits
• If f(x) is continuous on [𝑎, ∞), then
∞
𝑏
න f x dx = lim න 𝑓 𝑥 𝑑𝑥
𝑏→∞
𝑎
𝑎
• If f(x) is continuous on −∞, 𝑎 , then
𝑎
𝑎
න f x dx = lim න 𝑓 𝑥 𝑑𝑥
𝑏→−∞
−∞
𝑏
• Given any real number a, if f(x) is continuous on −∞, ∞ , then
∞
𝑎
∞
න f x dx = න 𝑓 𝑥 𝑑𝑥 + න 𝑓 𝑥 𝑑𝑥
−∞
−∞
𝑎
INTEGRAL CALCULUS I
PRACTICE PROBLEMS
∞ −x
22. Evaluate: ‫׬‬0 e
A. 0
B. 1
C. 2
D. 3
cos x + sin x dx
INTEGRAL CALCULUS I
PRACTICE PROBLEMS
23. Evaluate:
0
න 𝑥𝑒 2+3𝑥 𝑑𝑥
−∞
A. (e^2)/9
B. -9/(e^2)
C. –(e^2)/9
D. 9/(e^2)
ENGR. PETER BENJAMIN
OBIANO
INTEGRAL CALCULUS I
24. Evaluate:
∞
𝑧3
න 4
𝑑𝑧
𝑧 +1
−∞
A. -2.3
B. 0
C. 2.3
D. integral diverges
INTEGRAL CALCULUS I
IMPROPER INTEGRALS
Integrand with Infinite Discontinuities
• If f(x) is continuous on (𝑎, 𝑏], then
𝑏
𝑏
න f x dx = lim+ න 𝑓 𝑥 𝑑𝑥
𝑡→𝑎
𝑎
𝑡
• If f(x) is continuous on 𝑎, 𝑏 , then
𝑏
𝑡
න f x dx = lim− න 𝑓 𝑥 𝑑𝑥
𝑡→𝑏
𝑎
𝑎
• If f(x) is continuous on [𝑎, 𝑏] except on 𝑐 ∈ (𝑎, 𝑏), then
𝑏
𝑡
𝑏
න f x dx = lim− න 𝑓 𝑥 𝑑𝑥 + lim+ න 𝑓 𝑥 𝑑𝑥
𝑡→𝑐
𝑎
𝑡→𝑐
𝑎
𝑡
INTEGRAL CALCULUS I
25. Evaluate:
𝟏
1
𝑑𝑧
‫׬‬−𝟓
10+2𝑧
A. 4.35
B. -4.35
C. 0
D. integral diverges
INTEGRAL CALCULUS I
26. Evaluate:
4
1
‫׬‬0
2/5 dx
4−x
A. 2.829
B. 1.829
C. 3.829
D. 4.829
INTEGRAL CALCULUS 1
ENGR. PETER BENJAMIN B. OBIANO