integration problem (1 Viewer)

[ToO LaZy ^*]

i've just started integration last lesson and im a bit lost ??? ...here are my problems:

evaluate:
1) root(x) / root(1-x) [let u=2x+3]


2) 1 / (x^2 +1)^3/2

some help would be greatly appreciated

 

[CM_Tutor]

Originally posted by ToO LaZy ^*
evaluate:
1) root(x) / root(1-x) [let u=2x+3]

I'm not seeing the reason for the substitution u = 2x + 3, but the substitution u² = 1 - x works well...

Let I = int sqrt(x) / sqrt(1 - x) dx

Now, apply the reverse chain rule, and let u² = 1 - x. We see that 2u du = - dx, and x = 1 - u²

So, I = int [sqrt(1 - u²) / sqrt(u²] * (-2u) du = -2 * int u * sqrt(1 - u²) / u du
= -2 * int sqrt(1 - u² du

Now, again apply the reverse chain rule, this time with u = sin @, and so du = cos @ d@.

Hence, I = -2 * int sqrt(1 - sin²@) * cos @ d@ = -2 * int cos²@ d@
= -2 * (1 / 2) int 1 + cos2@ d@ = -[@ + (1 /2) * sin 2@] + C = -sin^-1u + u * sqrt(1 - u²) + C
= -sin^-1[sqrt(1 - x)] + sqrt(x - x²) + C

This is a mess ... I think there should be an easier way...

2) 1 / (x^2 +1)^3/2

This one is straight forward, once you realise it calls for the substitution x = tan @. The integral becomes
int 1 / sec @ d@ = sin @ + C = x / sqrt(1 + x²) + C

 

[ToO LaZy ^*]

thx heaps!!

edit...actually..im wrong, i copied down the wrong question..

[between 0.5-0] integrate:
root(x) / root(1-x).....then it tells you to use 'let x = sin^2 @'

 

[George W. Bush]

x = sin^2 @
dx = 2sin@cos@
When x = 1/2, @ = pi/4
x=0, @ = 0

Integral = sin@2sin@cos@/cos@ (as 0=<@=<pi/4) d@
= 2sin^2 @d@
cos2@ = 1 - 2sin^2@
so Integral = 1 - cos2@ d@
= @ - sin2@/2
= pi/4 - 1/2

 

[CrashOveride]

Hi Too Lazy,

What school do you goto?

It appears i may be near, judging from your location...


BTW, i was just curious as to why mathematical induction isn't a topic is any Maths textbook? We did it in year 11 for 3unit briefly and were asked harder induction questions in our 4unit Half-yearly. So what's the deal there ?

 

[George W. Bush]

I think it'd be in most 3u textbooks, as well as in the Harder 3U Maths section of 4u textbooks. At least, it is in my textbooks.

 

[ToO LaZy ^*]

thx bush!

 

[gress]

Originally posted by CrashOveride

BTW, i was just curious as to why mathematical induction isn't a topic is any Maths textbook? We did it in year 11 for 3unit briefly and were asked harder induction questions in our 4unit Half-yearly. So what's the deal there ?

I believe it's covered moreso in the 2unit topic 'Series and Sequences'.
Earlier this year I asked my tutor why Mathematical Induction wasn't in Jones/Couchman 3unit maths book and he said it was excluded as it was mainly covered in 2unit.

 

[CM_Tutor]

Originally posted by gress
I believe it's covered moreso in the 2unit topic 'Series and Sequences'.
Earlier this year I asked my tutor why Mathematical Induction wasn't in Jones/Couchman 3unit maths book and he said it was excluded as it was mainly covered in 2unit.

Sorry, but your tutor is wrong. Induction is a 3u topic, and is not covered by the 2u Jones and Couchman book - at least, it isn't in my copy!

The only coverage of induction by Jones and Couchman is in the 3unit book 1, chapter 14 (Sequences and Series) exercise 14.15. The 2u book 2 chapter on sequences and series stops at 14.14, and does not include the induction material.

Also, note that Jones and Couchman's coverage of induction in 3u, book 1, is woefully inadequate. It misses many important examples, and its inclusion with Sequences and Series shows a lack of appreciation for its significance. It is true that induction is often applied to series, but its application is much broader than this.

The coverage in 3u Fitzpatrick, chapter 23, is better - at least it has a more comprehensive coverage. The coverage in the appendix of brown Coroneos (3u course, yr 11), is better (IMO) than Fitzpatrick's, as it also covers examples of inequality and calculus induction, as well as having questions on factorial induction. I don't have Cambridge to hand, so I can't comment on the coverage, but I'd be surprised if ther was not a good induction section.

 

[CrashOveride]

Oh thanks for the clarification CM_TUTOR !

The thing is, we've only ever used Jones and Couchman 3unit book 2....i've seen book 1 like once in my life

And in book 2 there in no mention of induction....in patel's book for 4unit there is some brief inclusion of it, part of the 'harder 3unit' topic. But im a bit annoyed because i think our teacher has taught us in a really narrow way so we cant answer a whole variety of induction questions....

I wanted to have a look at it in a textbook, because i want to see how its introduced and so on.

 

[CM_Tutor]

CrashOveride, I have posted a variety of stuff on induction in the Extn 1 forum recently - if you go have a look, there are a number of examples of proofs, and also some broader examples.

 

[gress]

Originally posted by CM_Tutor
The only coverage of induction by Jones and Couchman is in the 3unit book 1, chapter 14 (Sequences and Series) exercise 14.15. The 2u book 2 chapter on sequences and series stops at 14.14, and does not include the induction material.

Also, note that Jones and Couchman's coverage of induction in 3u, book 1, is woefully inadequate. It misses many important examples, and its inclusion with Sequences and Series shows a lack of appreciation for its significance. It is true that induction is often applied to series, but its application is much broader than this.

The coverage in 3u Fitzpatrick, chapter 23, is better - at least it has a more comprehensive coverage. The coverage in the appendix of brown Coroneos (3u course, yr 11), is better (IMO) than Fitzpatrick's, as it also covers examples of inequality and calculus induction, as well as having questions on factorial induction. I don't have Cambridge to hand, so I can't comment on the coverage, but I'd be surprised if ther was not a good induction section.

Ahh wow, thanks for the reply CM_Tutor