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1. INTEGRAL CALCULUS:
A COMPREHENSIVE
OVERVIEW
Presented by Simbisai Chinhema
Sect 43 BI
Reg. No: 241FA14061

2. WHAT IS INTEGRAL CALCULUS?
• Integral calculus is the study of:
• - Antiderivatives (reversing differentiation).
• - Calculating areas under curves.
• Applications in real life such as in physics, engineering, and
economics.

3. WHY STUDY INTEGRAL
CALCULUS?
• - Solves practical problems like motion, area, and volume.
• - Provides tools for understanding rates of change.
• - Crucial for scientific and engineering advancements.

4. KEY CONCEPT: ANTIDERIVATIVE
• Integration reverses differentiation:
• Take for example!
• If F'(x) = f(x), then F(x) is the antiderivative of f(x). then
• f(x) = 2x F(x) = x^2 + C.
→

5. NOTATION AND TERMS
Here are some of the key notations and terms used in
integration:
- ∫…………. Integration symbol.
- dx………..Indicates the variable of integration.
C………… Constant of integration (indefinite integrals) .

6. POWER RULE FOR INTEGRATION
• The power rule is used for any value of x raised to another
value n as illustrated below:
• Formula: ∫xn
dx = x(n+1)
/(n+1) + C ( where n ≠ -1)
• Example:
∫x^3 dx = x4
/4 + C.

7. EXPONENTIAL AND
LOGARITHMIC RULES
• - Exponential: ∫ex
dx = ex
+ C.
• Example: ∫ e(3x)
dx = (1/3) * e(3x)
+ C
the integral of e^(3x) is (1/3) * e(3x)
+ C.
• - Logarithmic example: ∫(1/x) dx = log|x| + C.

8. TRIGONOMETRIC INTEGRALS
• Trigonometric integrals goes in hand with some fomulars used
for example
∫sin(x) dx = -cos(x) + C.
∫cos(x) dx = sin(x) + C.
∫tan(x) dx =sec(x)2
+ C.

9. INTEGRATION BY SUBSTITUTION
• Used to simplify complex integrals:
• Substitute u = g(x), then integrate in terms of u.

10. EXAMPLE: SUBSTITUTION
• Evaluate: ∫2x e^(x^2) dx
• Solution:
• - Let u = x^2 du = 2x dx.
→
• - ∫e^u du = e^u + C = e^(x^2) + C.

11. INTEGRATION BY PARTS
• Formula: ∫f(x)g(x) dv = f(x) ∫ g(x) dx- ∫f(x)’ * ∫ g(x) dx .
• Used when the integrand is a product of two functions.
• To chose the first function wisely there is need to use the ILATE
I…...Inverse function
L…...Logarithmic function
A……Arithmetic function
T……Trigonometrical function
E…….Exponential function

12. EXAMPLE: INTEGRATION BY
PARTS
• Evaluate: ∫x sin(x) dx
• Solution:
• - Let u = x, dv = sin(x) dx.
• - ∫x sin(x) dx = -x cos(x) + sin(x) + C.

13. DEFINITE INTEGRATION
• Calculates area under a curve between two points:
• ∫[a to b] f(x) dx = F(b) - F(a).

14. EXAMPLE: DEFINITE
INTEGRATION
• Evaluate: ∫[0 to 2] 3x^2 dx
• Solution:
• ∫3x^2 dx = x^3 [x^3] from 0 to 2 = 8.
→

15. APPLICATIONS: TANGENTS AND
NORMALS
• Find slope of tangent and normal lines:
• - Tangent: Slope = f'(x).
• - Normal: Slope = -1/f'(x).

16. EXAMPLE: TANGENT LINE
• Find the tangent to y = x^2 at x = 1.
• Solution:
• - Slope = 2x f'(1) = 2.
→
• - Equation: y - 1 = 2(x - 1).

17. APPLICATIONS: PHYSICS
• - Velocity: v(t) = ∫a(t) dt + C.
• - Position: x(t) = ∫v(t) dt + C.

18. EXAMPLE: MOTION
• Given a(t) = 2t, find v(t) and x(t).
• Solution:
• - v(t) = ∫2t dt = t^2 + C1.
• - x(t) = ∫(t^2) dt = t^3/3 + C2.

19. APPLICATIONS: ECONOMICS
• Used for:
• - Total cost/revenue calculations.
• - Consumer/producer surplus.

20. APPLICATIONS: ENGINEERING
• - Fluid flow analysis.
• - Structural behavior.
• - Heat transfer.

21. APPLICATIONS: PROBABILITY
• - Expected values in probability theory.
• - Variance calculations.

22. SUMMARY
• - Integration reverses differentiation.
• - Methods: Substitution, parts, partial fractions.
• - Applications span physics, engineering, and more.

23. THANK YOU!