Integration Help (1 Viewer)

[haboozin]

hey,
I have a question but before i post it..
I really find the way u guys solve integration in this forum weird, like i can solve almost all of the questions posted but never the way u guys do. Also its so hard to understand them with the poor formatting so please do your best to format the answer well.
Also tell me why u used the the 2 parts.

Use Integration by parts and the table of standard integrals to show that

Int SQRT(y² + 8) dy = 1/2y SQRT(y² + 8) + 4 ln(y + SQRT(y² + 8)) + c

 

[who_loves_maths]

i'll do the best i can with the formatting haboozin, but no guarantee that it'll sastify your expectation.

Show that: Int SQRT(y^2 + 8) dy = 1/2y SQRT(y^2 + 8) + 4 ln(y + SQRT(y^2 + 8)) + c using integration by parts...

Notation: i'm leaving out the 'dy' bits in Int[F(y)dy] for less confusion, ie.
Int[F(y)dy] becomes Int[F(y)]

let u = Sqrt(y^2 + 8) ---> du = y/Sqrt(y^2 +8)dy ; let dv =dy ---> y = v

ie. I = ySqrt(y^2 + 8) - Int[y^2/Sqrt(y^2 +8)] ; where Int[y^2/Sqrt(y^2 +8)] =Int[Sqrt(y^2 +8)] -8Int[1/Sqrt(y^2 +8)]

using the standard integrals: Int[1/Sqrt(y^2 +8)] = ln(y + Sqrt(y^2+8)) +C ;

hence, I = ySqrt(y^2 +8) +8ln(y +Sqrt(y^2 +8)) - Int[Sqrt(y^2 +8)] + C

but I = Int[Sqrt(y^2 +8)] ; therefore, 2I = ySqrt(y^2 +8) +8ln(y +Sqrt(y^2 +8)) +C

-----> I = (1/2)ySqrt(y^2 +8) + 4ln(y +Sqrt(y^2 +8)) +C = Int[Sqrt(y^2 +8)]


hope that helps, and was clear enough for you

 

[who_loves_maths]

haboozin,

in case you find that confusing or can't follow it, then i suggest you go and look through the examples in a book like Patel or Cambridge.

because, Int[(y^2 +8)dy] is the equivalent of 8Int[(Sec(x))^3 dx], where y =2sqrt2*Tan(x) ;

and (Sec(x))^3 is a 4unit integral that is demonstrated/taught in most 4unit textbooks...

 

[~ ReNcH ~]

Whoa....finally FinalFantasy was beaten to an integration question

 

[haboozin]

where Int[y^2/Sqrt(y^2 +8)] =Int[Sqrt(y^2 +8)] -8Int[1/Sqrt(y^2 +8)]

OK i did all that until this step... but i would have never thought of that step...

I dont think this quesiton will ever be in the hsc
without us first getting to do this step in like i.

 

[FinalFantasy]

Whoa....finally FinalFantasy was beaten to an integration question

hahaha, notin surprising about dat
heaps of ppl are heaps higher lvl den i am