any1 noe how to do this q? (2 Viewers)

[dumhsckid]

integrate
f (x) = 20x/(x^2+4) + 15

 

[A High Way Man]

its a substitution qn.

let u = x^2
du = 2x dx

so the integral becomes

int((10*du/(u+4)) + int(15 dx)

10 * int(du/u+4) + 15*int(dx)

10 * ln(u + 4) + 15x + C

10*ln(x^2+4) + 15x + C

 

[Slidey]

Another way, similar to above:

u=x^2+4
du=2x dx
Int 10/u du + Int 15 dx
= 10ln(x^2+4) + 15x + C

 

[Trebla]

Substitution is long and boring. It can be done without it.
∫[20x/(x² + 4) + 15] dx
= 10∫[2x/(x² + 4)] dx + 15∫dx
Whenever you see a derivative of the denominator in the numerator it's always ln(demoninator) so:
= 10ln(x² + 4) + 15x + c

 

[Slidey]

Substitution is long and boring. It can be done without it.
∫[20x/(x² + 4) + 15] dx
= 10∫[2x/(x² + 4)] dx + 15∫dx
Whenever you see a derivative of the denominator in the numerator it's always ln(demoninator) so:
= 10ln(x² + 4) + 15x + c

You do realise you just implicitly used substitution.

Not to mention that way is no quicker at all.

 

[Trebla]

You do realise you just implicitly used substitution.

Not to mention that way is no quicker at all.

How is it not quicker? A formal substitution involves the steps of differentiation and substituting it into the integral, then resubstituting it back after finding the primitive. However, it's virtually a one-liner without any formal substitution.

Also, 2 unit people are expected to know how to do these problems and there is no integration by substitution in the 2 unit course, so this is the preferred method.

 

[Slidey]

Fair point mate. I guess a better thing to say would be "substitution isn't slower".

 

[midifile]

Also, in 3 unit you should really only do substitutions when you are given the substitution. You will never be expected to work out what to substitute, that is only part of the 4 unit course, so if the question does not have a substitution there should be another way to do it (ie treblas method)

 

[Slidey]

Also, in 3 unit you should really only do substitutions when you are given the substitution. You will never be expected to work out what to substitute, that is only part of the 4 unit course, so if the question does not have a substitution there should be another way to do it (ie treblas method)

Supposedly. I've seen exam questions which don't give a substitution and are extremely hard (or impossible) without one (yes, in 3u). I highly recommend 3unit students learn when to make what substitution anyway, as it gets you into a good habit. Basically when you get an integration question in an exam, you can do it faster, and maybe even solve a hard integration question using an easy substitution (not supplied).

I know everybody gets told that you always get given the sub in 3u, but I wonder how much it is a general exam pattern compared to an actual syllabus specification.

 

[Iruka]

Just checked that Slidey.

In the syllabus, concerning 3u integration, it includes

"Methods of integration, including reduction to standard forms by very​

simple substitutions."

In the notes at the end of the syllabus it also states,

"In all examples the substitution is to be given."

 

[Slidey]

Good to hear.