Integration Questions

Ahm97sic SG

These questions are actually "O" level additional mathematics integration questions but these questions are more tricky.

Questions

(1)     Integrate [(x+1)^3+1]^3(x+1)^2
(2)     Integrate (x^5 + 1)/(x+1)
(3)     Integrate 2 sin^2x
(4)     Integrate sin^4x cosx
(5)     Integrate (1-sin2x)/(x-sin^2x)

Thank you for your kind attention.

Regards,

ahm97sic

eagle SG

Hi,

Is it ok if I add these questions to ExamWorld?

Thanks.

^tamago^ SG

I'll just do the first 3 questions for this.

Ahm97sic SG

I'll just do the first 3 questions for this.

Hi Tamago,

You don't have to expand out to integrate the first question. There is a shortcut to integrate it, the answer is very simplified. Hint: Use the reversal method.

Thank you for your kind attention.

Regards,

ahm97sic

eagle SG

these questions seems more like A level integration last time... Doesn't seem like the very standard O level

eagle SG

**(1)

**(4)

**(5)

Ahm97sic SG

Hi Eagle,

Basically, these questions use the reversal method but these questions are just more tricky.

"O" Level add maths also test on the use of reversal method.

Actual past "O" level add maths questions on the use of reversal method

(1) Differentiate 1/[(9 - 4x^2] and then integrate x/[(9 - 4x^2]^2

(2) Differentiate xsinx and then integrate xcosx

(3) Differentiate x^3lnx and then integrate x^2lnx

(4) .......

Indeed, with the addition of partial fraction and further trigonometry, more varieties of questions can be set on integration eg integrate (17x^2 +23x + 12)/[(3x+4)(x^2+4) in the new 2008 syllabus of Additional Mathematics.

Thank you for your kind attention.

Regards,

ahm97sic

PS : Hi Eagle, you have copied the first question incorrectly ie you have ommitted the power 3. However, based on the question you copied, your answer is correct. You have forgotten to type the dx in your answer for question 5 and you have made a mistake in your answer to question 5 ie when you differentiate your answer, you cannot get back (1-sin2x)/(x-sin^2x), not due to the constant c, it is the power of -2. When we differentiate it, the power of -2 becomes -3, not -1.

Hint : Use logarithm ln

^tamago^ SG

Hi Tamago,

You don't have to expand out to integrate the first question. There is a shortcut to integrate it, the answer is very simplified. Hint: Use the reversal method.

Thank you for your kind attention.

Regards,

ahm97sic

Q1.

Q5.

What's the purpose of this thread?

eagle SG

oops

sry... careless... I did everything mentally on MS word... Changed the answers on my blog liao

damn sleepy doing up ExamWorld

weewee SG

all use chain rule reversal method...

Ahm97sic SG

Q1.

Q5.

What's the purpose of this thread?

Hi Tamago and Eagle,

The purpose of these integration questions is let students who are doing Add Maths to know (and try to solve) of the more tricky questions especially with the addition of partial fraction, further trigonometry in the new syllabus that can led to more varieties of questions to be set in the "O" level exam.

Eagle and Tamago, both of you have kindly worked out the solutions to these questions for the benefits of the "O" level Add Maths students.

Indeed, the homework forum should not just help to do the homework for the students, the forum should also encourage more students to visit the forum by providing exam questions (with provided solutions), exam tips, types of questions and the different ways how a topic are being set in the PSLE, "N" Level, "O" level and "A" level. Currently, the forum is a bit messy, it should be organised into different sub-headings eg Lower Primary, Upper Primary, PSLE, lower secondary, upper secondary, "N" Level, "O" Level, "A" Level and others.

Students should be encouraged to post interesting (ie tough) questions that they encountered in their schools so that students from other schools should also be able to try to solve and benefit from these interesting questions.

Hope that my suggestions will be of help.

Thank you for your kind attention.

ahm97sic

PS : Hi Tamago, you have used the "A" level substitution method to do the integration but the "O" level add maths students have not learnt this method, they are only taught the reversal method to do the integration.

^tamago^ SG

Never heard of reversal method. Maybe you can share with us? Very Happy

eagle SG

We are not in control on the sub headings of the forum.

And not possible to have all questions of the same level in a single thread. This is because students might ask more about the questions, or we might be handling a few questions at a time. It will greatly confuse the students if we do that.

That's why I set up ExamWorld ;)

Ahm97sic SG

Never heard of reversal method. Maybe you can share with us? Very Happy

Hi Tamago,

Let's use an actual past "O" level question as an illustration of the reversal method.

Example

Differentiate 1/[(9 - 4x^2] and then integrate x/[(9 - 4x^2]^2

Let y = 1 / [(9 - 4x^2]

dy/dx = - 8x / [(9 - 4x^2]^2

To integrate x/[(9 - 4x^2]^2, we will try to make x/[(9 - 4x^2]^2 to be same as the dy/dx answer ie - 8x / [(9 - 4x^2]^2,

Integrate x / [(9 - 4x^2]^2

= -1/8 integrate - 8x / [(9 - 4x^2]^2

Without doing any integration, by reversal ie we know integrate dy/dx = y, so

= - 1/8 {1 / [(9 - 4x^2]}

Thank you for your kind attention.

Regards,

ahm97sic

PS : Hi Tamago, "O" Level add maths students have not learnt the "A" level substitution method to do the integration, they are only taught to use the reversal method to do the integration.

Using the reversal method to integrate, there is no need to have the constant c in the answers as it is an exact reversal of y.

eagle SG

ahm97sic

would like to invite you to join in at ExamWorld. :D

I can put an advertisement for your tuition services right under the subjects section, above the google search function. Currently looking for tuition teachers to help in posting and categorising for free advertisements.

Ahm97sic SG

oops

sry... careless... I did everything mentally on MS word... Changed the answers on my blog liao

damn sleepy doing up ExamWorld

Hi Eagle,

Using the reversal method to integrate, there is no need to have the constant c in the answers as it is an exact reversal of y.

Thank you for your kind attention.

Regards,

ahm97sic

eagle SG

Hi Eagle,

Using the reversal method to integrate, there is no need to have the constant c in the answers as it is an exact reversal of y.

Thank you for your kind attention.

Regards,

ahm97sic

I believe the constant c is still required.

It is only not required if it is a definite integral.

Correct me if I'm wrong. Thanks.

Ahm97sic SG

Hi Eagle,

Let's use a simplified example

y = 2x^3 + 5

dy/dx = 6x^2

If y is not given at first and we are asked to integrate 6x^2,

Integrate 6x^2

= (6x^3)/3

= 2x^3

There is no way to get the 5 in the original y, so there is a need to add the constant c to represent a possible constant in the original y, so

= 2x^3 + c

But when the question asks

Differentiate y = 2x^3 + 5 and integrate 7x^2

y = 2x^3 + 5

dy/dx = 6x^2

To integrate 7x^2, we try to make 7x^2 to be the same as the dy/dx answer ie 6x^2.

Integrate 7x^2

= 7/6 integrate {6x^2}

Without doing any integration, by reversal integrate dy/dx = y,

= 7/6 {2x^3 + 5}

Thus, the 5 in the original y is accounted for and so there is no need to have the constant c to represent a possible constant in the original y (as in the case of the indefinte integral).

Thank you for your kind attention.

Regards,

ahm97sic

eagle SG

Hi Eagle,

Let's use a simplified example

y = 2x^3 + 5

dy/dx = 6x^2

If y is not given at first and we are asked to integrate 6x^2,

Integrate 6x^2

= (6x^3)/3

= 2x^3

There is no way to get the 5 in the original y, so there is a need to add the constant c to represent a possible value in the original y.

But when the question asks

Differentiate y = 2x^3 + 5 and integrate 7x^2

y = 2x^3 + 5

dy/dx = 6x^2

To integrate 7x^2, we try to make 7x^2 to be the same as the dy/dx answer ie 6x^2.

Integrate 7x^2

= 7/6 integrate {6x^2}

Without doing any integration, by reversal integrate dy/dx = y,

= 7/6 {2x^3 + 5}

Thus, the 5 in the original y is accounted for and so there is no need to have the constant c to represent a possible value in the original y (as in the case of the indefinte integral).

Thank you for your kind attention.

Regards,

ahm97sic

The reason for adding + c is because it is a general case.

If you differentiate 7/6 {2x^3 + 5}, you get 7x^2

but if you differentiate 7/6 {2x^3 + 5} + c, you still get 7x^2

This is because c can be 7/6 * 5 or any other value.

That's the reason for adding + c for indefinite integrals.
Not sure if you are referring to the questions in the first post... or other questions...

Even then, for questions that ask for differentiate, then integrate... a +c is still required. This is because the first part of the question serves to guide the student to be able to integrate in the 2nd part of the question. I don't really think they are linked.