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# Hit me with a difficult P3 or P1 question watch

1. Funny enough, but I have run out of exam questions. I am an A2 student, so I am resitting P1, as I got a D last year.

Please could someone give me a question I can solve, Id like to try a hard P1 one. Others can solve them too, then I can compare and see If im right.

Hints - I dont like trigonometry. But I need practice in this area.
2. Integrate lnx dx.
3. (Original post by Bhaal85)
Integrate lnx dx.
Can I have a go too?

u = lnx
du/dx = 1/x

dv/dx = 1
v = x

xlnx - Integral of x/x = xlnx - x + C
4. (Original post by Bhaal85)
Integrate lnx dx.
Is that in P1?
5. Here's a difficult one for anybody. It can be done using P3 concepts (in fact even if you only know P2), but I doubt it will come up in the exam:

Prove that,

tan^(-1) [(1+x) / (1-x)] = pi/4 + tan^-1 (x)

(in words, prove that tan inverse of [...] is equal to pi/4 + tan inverse of 'x')
6. (Original post by Bhaal85)
Integrate lnx dx.

Oh I can do this.

You use parts let U = lnx and let dv/dx = 1
du/dx = 1/x {1 .dx = x = v

uv - {v.du/dx

xlnx - {x/x

answer is = xlnx - x + c
7. (Original post by Jonny W)
Is that in P1?
no.....they tell you integral of 1/x 'can't be done' when you're doing P1, so we wouldn't be required ot know what 'ln' is
8. If you want more practice on the "ln(x) = 1*ln(x)" type of integration by parts, try finding

(int) arcsin(x) dx,
(int) arccos(x) dx,
(int) arctan(x) dx.
9. (Original post by Jonny W)
If you want more practice on the "ln(x) = 1*ln(x)" type of integration by parts, try finding

(int) arcsin(x) dx,
(int) arccos(x) dx,
(int) arctan(x) dx.
thats P5 for us
10. (Original post by Katie Heskins)
thats P5 for us

I dont do P5.

In P3, for the integration questions that need solving by substitution, is the substitution given all the time???
11. Using tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B)),

tan(pi/4 + arctan(x))
= (tan(pi/4) + x) / (1 - tan(pi/4) x)
= (1 + x) / (1 - x).

Now we want to apply arctan to each side. That would give the result - if it were true that arctan(tan(y)) = y for all y. Unfortunately, arctan(tan(y)) = y only holds for y with -pi/2 < y < pi/2.

So we have to make the extra assumption that -3pi/4 < arctan(x) < pi/4, ie, that x < 1.

So: arctan[(1 + x) / (1 - x)] = pi/4 + arctan(x) for all x < 1.
Attached Images

12. (Original post by BlueAngel)
I dont do P5.

In P3, for the integration questions that need solving by substitution, is the substitution given all the time???
Maybe not in simple cases such as

(int) u/(u-1)^2 du.
13. Integrate sin^2x dx. i.e (sinx)(sinx) dx
14. Integrate:

1/[x(lnx)^1/2] dx betweeen e^2 and e.
15. (Original post by Katie Heskins)
thats P5 for us
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.
16. (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.
What board is that? For Edexcel, you don't need to know anything about calculus with inverse trig functions in P3.
17. (Original post by Nylex)
What board is that? For Edexcel, you don't need to know anything about calculus with inverse trig functions in P3.
Nor with OCR, and we are told what to use in the substitution.
18. (Original post by Bhaal85)
Nor with OCR, and we are told what to use in the substitution.
Same with Edexcel P3 and substitutions, IIRC.
19. (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.
Is the first one 1/cos(arcsinx)?
20. (Original post by Nylex)
What board is that? For Edexcel, you don't need to know anything about calculus with inverse trig functions in P3.
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).

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