Maple Assignment ALL

WORKSHEET 1

QUESTION 1

Find a substitution u=u(x) that reduces the integral ∫ x /√(2 x¿+3)¿ dx to an integral of a polynomial in terms of u.

We choose to substitute (2x+3) in terms of u as it simplifies the denominator.

However, even though the equation is simpler, it is not simple enough. As such, √ (2 x+3) is

chosen to simplify the equation further. This way, the whole equation is easier to integrate.

QUESTION 2

Find a substitution for each of the following two integrals which reduce them to the same easy integral in terms of u.

Question 2(a)

We choose to substitute (x+3) with u to simplify the equation.

As the equation is still a bit hard to integrate, we choose to substitute √ (x+3) with u. This

further simplifies the equation to make it much easier to integrate.

Question 2(b)

As the equation is quite complicated, we choose to substitute x with u.

However, this substitution did not work as it did not simplify the equation. Instead, another

substitution is chosen. √ x is chosen to be substituted as u. This makes the equation very simple.

WORKSHEET 2

QUESTION 1

Find the long division ofw 1w 2

.

>

Define the polynomial first;

Then find the quotient (quo) and remainder (rem);

Lastly, use quo, rem and w2 to find the final form of quotient w1/w2;

QUESTION 2

>

a) Firstly, define r1;

Find directly the indefinite integral by using MAPLE 12 software.

Find the partial fraction decomposition. Since the degree of x in numerator r1 is lower than the

denominator of r1, thus long division method is not required.

Factorize the denominator of r1;

b) Define r2;

Integrate r2 directly using MAPLE 12;

Find the partial fraction decomposition without using long division method as the degree of x in numerator is lower than denominator.

;

Factorized r2;

Do you see the relationship between the three?

Yes, there is a relationship between the three equations because to find the partial fraction decomposition of r1 and r2, we have to first factor the denominator. The integration would be easier by integrating the partial fraction decomposition.

QUESTION 3

Try to factor the polynomial w=-4+10x+6x2-14x3 using the simple “factor” command. What happens?

>

It does not factorize.

Use the appropriate command to obtain a floating factorization of the polynomial w.

>

The above method is used because the function contains irrational roots. So, Maple solves this by

either factorization or partial fraction decomposition.

Consider the rational function:

>

QUESTION 4(a)

>

QUESTION 4(b)

QUESTION 4(c)

>

QUESTION 4(d)

>

For questions 4(c) and 4(d), a different method is used because the functions need to be

converted to partial fractions before they are integrated.

WORKSHEET 3

QUESTION 1

METHOD 1

The above method integrates the trigonometric function by expanding sec6x. However, the method is too long and complicated and so another method is chosen.

METHOD 2

Method 2 simplifies the function by substituting sec2x to (1+tan2x). The function is then expanded and makes it easier to integrate the overall function.

METHOD 3

Method 3 is the simplest amongst the three methods. Similarly like Method 2, sec2x is substituted with (1+tan2x). However, to simplify the function, tan x is substituted with u and integrated.

QUESTION 2

METHOD 1

The above method integrates the trigonometric function by expanding sec2x. However, the method is too long and complicated and so another method is chosen.

METHOD 2

Method 2 simplifies the function by substituting sec2x to (1+tan2x). The function is then expanded and makes it easier to integrate the overall function.

METHOD 3

Method 3 is the simplest amongst the three methods. Similarly like Method 2, sec2x is substituted with (1+tan2x). However, to simplify the function, tan x is substituted with u and integrated.

QUESTION 3

METHOD 1

The above method integrates the trigonometric function by expanding cot24x. However, the method is too long and complicated and so another method is chosen.

METHOD 2

Method 2 simplifies the function by substituting csc2x to (1+cot2x). The function is then expanded and makes it easier to integrate the overall function.

METHOD 3

restart

Method 3 is the simplest amongst the three methods. Similarly like Method 2, csc2x is substituted with (1+cot2x). However, to simplify the function, tan x is substituted with u and integrated.