How do you derive the second part of the equation from the first part given the following
sin P + sin Q = 2 sin 0.5(P+Q)cos0.5(P-Q) (1)
sinP cos Q = 0.5 [sin(P+Q) + sin (P-Q)] (2)
the first equation is given in MF15 (formula list) but I am not sure how I can manipulate it to get the 2nd equation. Need help urgently on this, have a test tmr.
Also, how do i integrate this?
Question: 1 / x ln x between e^2 and e^1
Ans is 0.693.
Thanks.
Hi,
Let A = 0.5(P + Q) and B = 0.5(P - Q).
Then A + B = P and A - B = Q.
Equation (1) can then be expressed as the form of equation (2).
The expression
1 / (x lnx)
can be rewritten as
(1/x) / (ln x),
which looks like f'(x) / f(x).
Thanks.
Cheers,
Wen Shih
Hi,
Some expressions may not be straightforward to be integrated, such as:
1. 1 / (cos 2x - 1);
2. 1 / [ sqrt(x) - sqrt(x - 1) ].
Thanks.
Cheers,
Wen Shih
Dear Audi,
>> How do i integrate
>> 1 / x ln x between e^2 and e^1
This question is very similar to question 17(a) from Miscellaneous Exercise 19 (p. 422) from Ho Soo Thong and Khor Nyak Hiong's Panpac Additional Mathematics. Please refer to http://ascklee.org/Stuff.html for the complete solution.
Sincerely,
ascklee
Dear ascklee,
Welcome! Do continue to contribute your expertise, given your rich teaching experience. Many of us will benefit from your sharing :)
Cheers,
Wen Shih
Originally posted by wee_ws:Dear ascklee,
Welcome! Do continue to contribute your expertise, given your rich teaching experience. Many of us will benefit from your sharing :)
Cheers,
Wen Shih
Pro!
Originally posted by wee_ws:Hi,
Some expressions may not be straightforward to be integrated, such as:
1. 1 / (cos 2x - 1);
2. 1 / [ sqrt(x) - sqrt(x - 1) ].
Thanks.
Cheers,
Wen Shih
These are nice questions :)
Q1 requires strength in trignometric identities, which sadly isn't present in most A level students nowadays due to the removal of Trigo as a main topic in the H2 syllabus. It's still omnipresent, but not well focussed.
Q2 requires rationalising, and involves knowing how to properly apply secondary school work of (a+b)(a-b).
To me, integration at H2 syllabus can be divided into 4 different types
1) Basic integration, very direct type
2) f'(x) vs f(x) type
3) By Substitution
4) By Parts
For number (3), the substitution is usually given.
As a strategy, I would advise my students to go down the list of 1, 2 and 4 (for H2) since this is in order of ease of integrating. If you cannot use 1, you move on to 2, and if still cannot, move on to 4. But as you move on to (4), do try to take note if (2) still applies.
Following this fixed strategy ensures your mind is geared towards trying to identify which is the correct method to use instead of getting bogged down by not even being able to apply what is needed.
Finally, remember that in exams, to check for your answer for this integration question, you do not need to redo it. Merely key the definite integral in your calculator, and compare your answer.
You do not need 100% to get A; you just need a high probability of having little careless mistakes on questions you have trained yourself to do.
Hi eagle,
Thanks for sharing your strategies that will help many students excel :)
To add further:
(a) Many direct integration questions require the recall of results like the integral of f'(x) / f(x), the integral of f'(x) / sqrt(f(x)), the integral of f'(x) [ f(x) ]^n, the integral of (ax + b) / sqrt(px^2 + qx + r), the integral of (ax + b) / (px^2 + qx + r), etc.. Completing the square is an important skill at times.
(b) One may encounter cases where long division and breaking into partial fractions are necessary, e.g., integrate (x^3 + 1)/(x^2 - 4).
(c) One may encounter cases (typically in school exam questions) where the substitution method is applied, then followed by the steps in (b). Sometimes, one has to do integration by parts after making use of the substitution method.
Thanks.
Cheers,
Wen Shih
We have a (d)
One may encounter cases too where one is tasked to do some initial differentiation or integration in part a, then an integration by parts in part b.
In those cases, LIATE may not apply at times.
Hi,
One may refer to N05/P1/Q3 as an example, where one has to split the expression x^3 cos(x^2) into x^2 and x cos (x^2). Thanks.
Cheers,
Wen Shih