integral of (cos x)^3

integral of (cos x)^3

the (cosx)^3 is split into (cos²x )cosx

the cos²x is written as 1 - sin²x (using trigonometry formulae)

then the substitution t = sinx , dt=cosx dx is used

more explanation is given below


click here for integration formulae


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integral of cosx / [ 1- cos²x]

using 1- cos²x = sin²x

∫ { cosx / [ 1- cos²x] } dx = ∫ { cosx / [sin²x] } dx

= ∫ {1 / sin x} {cos x / sin x} dx

= ∫ cosec x cot x dx

= - cosec x + c


click here for integration formulae

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solving a quadratic equation by completing the square

solving a quadratic equation by completing the square

 solve 2x² -4x -7 = 0 

 the equation can be written as 2x² -4x =7 

 dividing by 2 ( coefficient of x² )

 x² - 2 x = (7/2) 

note that coefficient of x is -2, take half of it that is -1,

 square that and add the square to both sides,

 use that to complete the square on the LHS x² - 2 x + 1² = (7/2) + 1²

 

 (x-1)² = (9/2) 

 

 take square root x -1 = ± ( 3 / √2 )

 

 x = 1 ± ( 3 / √2 ) 

 

 

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question on normal distribution

if the mean mark is 75 with a sd of 8 marks, what should you score to be in the top 25% .


z = (x - 75) / 8 is the standard normal variate

we have to find k so that

P[X > k ] = 0.25

P[ z > (k-75)/8 ] = 0.25


refer standard normal tables (note that probability is given)

(k-75)/8 = 0.675 (approximately)

k = 80.4 marks approximately


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buying tax saver elss mutual funds

It seems that the direct tax code implementation will only be from the next year onwards.
So the concept of income tax breaks for investments in elss mutual funds seems to be still good for this financial year.
I recently bought some units of hdfc tax saver and sbi magnum tax gain mutual funds.
I went to the cams outlet at pattom and gave a copy of the kyc acknowledgement along with the cheque and application form as it was a new folio for the hdfc mutual fund.

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buying a sewing machine

My wife wanted to buy a sewing machine for some time. We went to look for a sewing machine at a shop near the palayam market. There were many types of lower cost machines ranging from small tabletop machines (running on electricity) costing around 3000 rupees to 4000 rupees and manual sewing machines starting from around 5200 rupees for usha sewing machines , meritt sewing machines , singer sewing machines and so on. There were also some electrical sewing machines costing around 7000 rupees to 8000 rupees with a mixer (mixie ) as free gift.

My wife wanted to buy a manually operated singer sewing machine. The shop keeper told us that some of the internal parts of that particular sewing machine could be easily accessed by the lay user. The singer sewing machine cost around Rs.5800 /-. We used an autorickshaw to take it from the shop. The sewing machine came with some spare bobbins, screw drivers, some lubrication oil, washers etc.

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