It is cool, but wouldn't be required for anything easier than FP2.

Have you done FP2 out of curiosity? Because you seemed to accept my method pretty well.

Reply 12

SamKeene

Original post by morgan8002

It is cool, but wouldn't be required for anything easier than FP2.

Have you done FP2 out of curiosity? Because you seemed to accept my method pretty well.

Yeah (and FP3/4).

Speaking of awesome tricks, to return the favor (you might know already), ill share a trick I saw recently.

You can avoid using an integration by parts on:

\int \! e^x\cos{3x} \, \mathrm{d}x

And similar by noting this integrand is the real part of

e^x(\cos{3x}+i\sin{3x})

which can be rewritten as

e^xe^{3ix}

e^{1+3ix}

. Thus:

\int \! e^{1+3ix} \, \mathrm{d}x = \frac{1}{1+3i}e^{(1+3i)x}+k

But

\frac{1}{1+3i} = \frac{1}{(1+3i)}\frac{(1-3i)}{(1-3i)}=\frac{1-3i}{10}

Hence:

\frac{1}{1+3i}e^{(1+3i)x}=\frac{1}{10}e^x(1-3i)(\cos{3x}+i\sin{3x})

The required integral is the real part of:

\int \! e^{(1+3i)x} \, \mathrm{d}x

-->

\int \! e^x\cos{3x} \, \mathrm{d}x =\frac{1}{10}e^x (\cos{3x}+3\sin{3x}) + k

(edited 9 years ago)

Reply 13

username1560589

Original post by SamKeene

Yeah (and FP3/4).

Speaking of awesome tricks, to return the favor (you might know already), ill share a trick I saw recently.

You can avoid using an integration by parts on:

\int \! e^x\cos{3x} \, \mathrm{d}x

And similar by noting this integrand is the real part of

e^x(\cos{3x}+i\sin{3x})

which can be rewritten as

e^xe^{3ix}

e^{1+3ix}

. Thus:

\int \! e^{1+3ix} \, \mathrm{d}x = \frac{1}{1+3i}e^{(1+3i)x}+k

But

\frac{1}{1+3i} = \frac{1}{(1+3i)}\frac{(1-3i)}{(1-3i)}=\frac{1-3i}{10}

Hence:

\frac{1}{1+3i}e^{(1+3i)x}=\frac{1}{10}e^x(1-3i)(\cos{3x}+i\sin{3x})

The required integral is the real part of:

\int \! e^{(1+3i)x} \, \mathrm{d}x

-->

\int \! e^x\cos{3x} \, \mathrm{d}x =\frac{1}{10}e^x (\cos{3x}+3\sin{3x})

I've done FP2 and FP3 and am taking in May/June. I am doing FP4 currently.

That is a good trick. Thanks, I hadn't seen that before. Double parts can be tedious.
You missed a bracket pair on two of the lines, but you seem to have corrected later.

(edited 9 years ago)

Reply 14

SamKeene

Original post by morgan8002

I've done FP2 and FP3 and am taking in May/June. I am doing FP4 currently.

That is a good trick. Thanks, I hadn't seen that before.
You missed a bracket pair on two of the lines, but you seem to have corrected later.

Missed brackets? :redface:

Where

FP4 is pretty awesome, if I remember its just Matrices and Vectors which are awesome with the synergies they have with each others and other areas of maths.

Reply 15

username1560589

Original post by SamKeene

Missed brackets? :redface:

Where

FP4 is pretty awesome, if I remember its just Matrices and Vectors which are awesome with the synergies they have with each others and other areas of maths.

It should be

(1+3i)x

, but in two places you wrote

1+3ix

.

Yeah it's good. I'm still trying to get my head around some areas though.

Reply 16

SamKeene

Original post by morgan8002

It should be

(1+3i)x

, but in two places you wrote

1+3ix

.

Yeah it's good. I'm still trying to get my head around some areas though.

What kind of stuff have you covered so far/hard to get your head around?

If you want a book which really covers the FP1-FP4 vectors/matrices stuff well, I suggest Bostock and Chandlers Further Pure Mathematics. It lends itself to a deeper understanding and also explains the relevance of things like orthogonal matrix/diagonalization etc (which most A level textbooks won't).

Reply 17

TeeEm

Original post by asadmoosvi

How do I solve the following equation? I need help in simplifying the roots.

I need help in simplifying the roots.

the square root requires clever manipulations

√[8 +8i√3]
=√[8 +2.4i√3]
=√[12-4 +2.4i√3]
=√[(2√3)² +(2i)² +2.2i.2√3]
=√[2i+2√3]²
=2i+2√3

Reply 18

username1560589

Original post by SamKeene

I didn't start that long ago. I did a few days before Christmas and then most of last week as well. I've done the first 5 chapters out of 7 of the AQA textbook, but so far I've skipped over the areas I've struggled with, with the intention of going over it later.
So far it's factorising determinants and vector equations of lines and planes.
I think the vector equations just needs going over properly once for a few hours, because there are just so many forms etc.. I plan to do that tomorrow after college and get it sorted out properly.
As for factorising determinants, I don't think I will get it from the textbook, so I'm going to have a look for something online or something.

Thanks for the suggestion. I will look into it. FP1 has very little matrices and no vectors, it's just FP4 where they put it all.