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Hit me with a difficult P3 or P1 question watch

1. (Original post by Jonny W)
P3 students should know:
- how to differentiate arcsin, arccos and arctan;
- how to integrate x/sqrt(1 - x^2) and x/(1 + x^2);
- how to integrate ln(x) by parts.

Hence they could reasonably be asked to integrate arcsin, arccos and arctan. That said, those problems are probably too hard for a P3 exam.
woah, which exam board is this? That seems so harsh.
On MEI, for P3 you need to be able to do integration by parts and substitution. Then calculus on trig functions ie sin, cos, tan and powers of them
2. (Original post by Jonny W)
The Edexcel P3 specification (http://www.edexcel.org.uk/VirtualContent/25626.pdf) includes "the use of dx/dy = 1 / (dy/dx)".

y = sin(x)
x = arcsin(y)
dx/dy = 1 / (dy/dx) = 1 / cos(x) = 1 / sqrt(1 - x^2).
Yeah, but it doesn't say anything differentiating about inverse functions. That's listed under the P5 specification.
3. (Original post by Katie Heskins)
woah, which exam board is this? That seems so harsh.
Integrating inverse trig functions is probably too hard for P3 exams, but OK for homework.
4. (Original post by Jonny W)
Integrating inverse trig functions is probably too hard for P3 exams, but OK for homework.
It would not be in the exam. You are right that the calculus required to deal with inverse trig functions is available in P3, but it is not part of the syllabus.
5. Integrate (4 + x^4)^-1 dx

Hint: x^4 + 4 = (x^2 + 2x + 2)(x^2 - 2x +2)
6. (Original post by fishpaste)
Integrate (4 + x^4)^-1 dx
ln|4 + x^4| / 4x³?
7. (Original post by shift3)
ln|4 + x^4| / 4x³?
No. It looks like you used a substitution of u = 4 + x^4 and then didn't integrate the 4x^3 which resulted.
8. (Original post by shift3)
ln|4 + x^4| / 4x³?
no.
9. (Original post by IntegralAnomaly)
no.
further clues ?

substitution?
10. (Original post by kikzen)
further clues ?

substitution?
work on fishpaste's hint (x^4+4)=(x^2 + 2x + 2)(x^2 - 2x +2).
So expess (x^4+4)^-1 in the form of (Ax+B)/(x^2+2x+2) + (Cx+D)/(x^2-2x+2).(and remember the differential of arctanx)
11. (Original post by kikzen)
further clues ?

substitution?
How might you use the clue given up there?
12. How about 4x + (1/5)x^5?
13. (Original post by IntegralAnomaly)
work on fishpaste's hint (x^4+4)=(x^2 + 2x + 2)(x^2 - 2x +2).
So expess (x^4+4)^-1 in the form of (Ax+B)/(x^2+2x+2) + (Cx+D)/(x^2-2x+2).(and remember the differential of arctanx)
Woah.
14. (Original post by fishpaste)
Integrate (4 + x^4)^-1 dx

Hint: x^4 + 4 = (x^2 + 2x + 2)(x^2 - 2x +2)
I get this nasty thing:

[ ln(x^2 + 2x + 2) - ln(x^2 - 2x + 2) + 2arctan(x+1) + 2arctan(x-1) ]/16 + C
15. (Original post by mikesgt2)
I get this nasty thing:

[ ln(x^2 + 2x + 2) - ln(x^2 - 2x + 2) + 2arctan(x+1) + 2arctan(x-1) ]/16 + C
thats what i got to.
16. (Original post by IntegralAnomaly)
thats what i got to.
Its a bit of a slog, dealing with a lot of partial fractions and algebra.
17. Some difficult P1, as I can't ask questions I don't understand!

1. A curve has the equation:

x = 4y^2 - 9
Find the area between the curve and the line x = 1

Find the minimum value of x, and the value of y that gives this.

When y = 17, and c is set to 9, find the equations of the normal to the curve and find where it meets the x axis.
18. Much respect to you for getting that far. Well done.
19. I have NEVER answered a question about arctanx or arccosx or all of those. I swear to god if those come up i am buggered.
20. (Original post by *dave*)
I have NEVER answered a question about arctanx or arccosx or all of those. I swear to god if those come up i am buggered.
calculus involving inverse trig is p5 for edexcel.

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