2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

TIF 21101APPLIED MATH 1

(MATEMATIKA TERAPAN 1)

Week 13Anti Derivative and Integral

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Let’s ReviewDifferentiation 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

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].”

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.

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Anti Derivative and Integration

4.

5.

6.

7.

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Anti Derivative and Integration

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

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 :

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Anti Derivative and Integration

Example 2 :

Find

Ans :

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Anti Derivative and Integration

Partial Integration

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 :

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.

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Anti Derivative and Integration

Examples 3:

Find

Answer :

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Anti Derivative and Integration

Example 4:

Answer :

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Anti Derivative and Integration

Definite IntegrationSo 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 :

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Anti Derivative and Integration

Example 5:

Find

Answer :

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

Anti Derivative and Integration