Indefinite Integrals
Bridge Course, August 2012

Definition

Let f(x) be a function. If F(x) is another function such that its derivative is equal to f(x), that is,

                                                                        

then F(x) is called the primitive function. Note that primitive function is not unique. For example,

                                                  

All  are primitive function of 2x.

In general, if F(x) is a primitive function of f(x) and C is a constant, then

                                                            

If F(x) is one of the primitive function of f(x), then we have

                              

This process of solving for F(x) from a giving f(x) is called indefinite integration and its opposite function is called differentiation, which is the process of finding a derivative.

 

Properties

Examples                                                                                                           MAXIMA code

1. Evaluate                                                                                                        integrate(2/x, x)

   

2. Evaluate                                                                              integrate((x + 1) ^ 2/x, x)

   

 

Method of Substitution

Let u = (x) be a differentiable function and . Then, 

                                                    

Note that we can simply write as: 

 

Examples

1. Evaluate                                                                                  integrate(cos(3 ∗ x), x)

Let u = 3x and du = 3dx

Hence, 

                                        

2. Evaluate                                                              integrate(x ^ 2 ∗ (2 ∗ x ^ 3 + 5) ^ 5, x)

Let u = 2 − x and du = −dx

Hence,

                                             

3. Evaluate                                            integrate(x ^ 2 ∗ (2 ∗ x ^ 3 + 5) ^ 5, x)

Let u = 2 + 3 and du = 6dx

Hence, 

 

Exercises

Evaluate the following integrals by method of substitution

1.                                                                             integrate(x ^ 2 ∗ sqrt(x − 3), x)

2.                                      integrate((x + 1) ∗ sin(x ^ 2 + 2 ∗ x + 1), x)

3.                                                               integrate(%e ^ x ∗ (1 − %e ^ x) ^ (−1/4), x)

4.                                                                 integrate(2/sqrt(x) ∗ %e ^ (−sqrt(x)), x)

5.                                                                             integrate(1/sqrt(a ^ 2 − x ^ 2), x)

6.                                                                                      integrate(1/(a ^ 2 + x ^ 2), x)

 

Integration of Rational Functions

A rational function  is a quotient of two polynomials P(x) and Q(x).When integrating the rational function, the usual technique we use is to decompose the rational function into partial fractions.

Note that: We have to ensure f(x) is a proper fraction(that is degree of P(x) must be less that Q(x)) before we decompose f(x) into partial fractions.

 

Some transformation of partial fractions

                    

 

Examples

1. Evaluate                                                                         integrate(1/(x ^ 2 − a ^ 2), x)

     First, we have to decompose it into partial fractions, 

     and we let 

     By solving, we get .                                       partfrac(1/(x ^ 2 − a ^ 2), x)

     Then, 

     

 

2. Evaluate                             integrate((4 ∗ x ^ 2 + x + 12)/(x ∗ (x ^ 2 + 4)), x)

     First, decompose into partial fractions, we let 

     By solving, we get      partfrac((4 ∗ x ^ 2+x+12)/(x ∗ (x ^ 2+4)), x)

     Then,

     

 

3. Evaluate                                               integrate((x ^ 2)/(x ^ 2 − 2 ∗ x + 1), x)

     Since , we let 

     By solving, we get partfrac((x ^ 2)/(x ^ 2−2 ∗ x+1), x)

     Then,

     

 

Exercises

Evaluate the following integrals

1.                                                                  integrate(x/(2 ∗ x ^ 2 + x − 3), x)

2.                                                                                                find it yourself

3.                                                                                             find it yourself

4.                                                                                      find it yourself

5.                                                                                          find it yourself

 

Integration by parts

Let u(x) and v(x) be two real-valued functions with continuous first derivatives. Then

                                                                 

The formula is simple so let’s get some illustration.

 

Examples

1. Evaluate                                                                            integrate(x ^ 2 ∗ %e ^ x, x)

     

2. Evaluate                                                                       integrate(x ∗ sin(2 ∗ x), x)

     

3. Evaluate                                                  integrate(x ^ n ∗ log(x), x)

     

4. Evaluate                                                                   integrate(cos(x) ∗ %e ^ x, x)

     

     Hence, we have

     

     where C and C' are constants.

 

Exercises

Evaluate the following integrals

1.                                                                                 integrate(x ∗ %e ^ (2 ∗ x), x)

2.                                                                                              find it yourself

3.                                                                                                find it yourself

4.                                                                                                       find it yourself

5.                                                                                                find it yourself