This document discusses anti-derivatives and integration. It begins by defining anti-derivatives as the process of determining the function from a given derivative, and defines integration as determining the anti-derivative. It then covers basic properties of integration, methods for evaluating integrals including substitution and integration by parts, and defines definite integration over an interval. Examples are provided to illustrate each concept.

1. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
TIF 21101
APPLIED MATH 1
(MATEMATIKA TERAPAN 1)
Week 13
Anti Derivative and Integral

2. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Let’s Review
Differentiation is the process of determining the derivative
from a given function.
Integration is the process of determining the anti-
derivative from a given function.
Thus, this implies that…
the derivative f’(x) is found by the differentiation of function f (x)
the anti-derivative F(x) is found by the integration of function f’(x)
Anti Derivative and Integration

3. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Anti Derivative and Integration
∫ (something) dx means “the antiderivative of (something)
with respect to x.”
This is how we say this aloud. The symbols ∫ dx are again an operator;
they are telling you to do something. Again, they go away after you do
what it tells you to do. This operator takes a single input and gives you
a family of functions as an output.
is the “definite integral of the function f(x) on the
interval [a, b].”

4. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Anti Derivative and Integration
Basic Properties
If f (x) and g (x) are functions (the anti
derivative exists), k and n are any real
numbers and c is constant, then:
1.
2.
3.

5. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Anti Derivative and Integration
4.
5.
6.
7.

6. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Anti Derivative and Integration

7. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Anti Derivative and Integration
Subtitution Method for Integration
To evaluate an antiderivative ∫ f(x) dx, it is
often useful to replace x with a new variable
u by means of a substitution x = g(u) and
dx = g’(u) du. Therefore, the equation
becomes:
∫ f(x) dx = ∫ f(g(u)) g’(u) du

8. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Anti Derivative and Integration
Example 1:
Evaluate
Answer :
Assume u = x+3, thus x = u – 3
dx/du = 1  dx = du. Going back to the
problem will give :

9. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Anti Derivative and Integration
Example 2 :
Find
Ans :

10. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Anti Derivative and Integration
Partial Integration

11. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Anti Derivative and Integration
By integration in respect to x to the both side of equation, it will give
Reforming the equation will give :

12. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Anti Derivative and Integration
When there is a problem that needs an integration,
the given integral must be separated into two
parts, one part being U and the other part, together
with dx, being dV.
For this reason, integration used above is called
integration by parts.
Two general rules can be stated:
1.The part selected as dv must be readily integrable.
2.∫ Vdu must not be more complex than ∫ Udv.

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M.Eng
Matematika Terapan 1
Anti Derivative and Integration
Examples 3:
Find
Answer :

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M.Eng
Matematika Terapan 1
Anti Derivative and Integration
Example 4:
Answer :

15. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Anti Derivative and Integration
Definite Integration
So far, the integral we have learned is the
indefinite integration. Let’s get on the definite one.
Basically, it is just the same with the previous
ones. Only, in this matter, we will put limitation on
it with [a,b].
If the function F(x) is the anti derivative to f(x)
which is continuous on [a,b], thus we can write the
formula into :

16. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Anti Derivative and Integration
Example 5:
Find
Answer :

17. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Anti Derivative and Integration