Mathematics: Post your doubts here!

leadingguy

leadingguy

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shanky631 said:
View attachment 14874

when x=4sinu

plz can someone help me. thanks in advance.:)
Click to expand...





x = 4sinu so dx= 4cosu.du

replace the x in eq. with 4sinu


√(16 - (4sinu)^2) .dx

√(16 - (4sinu)^2).4cosu.du as dx = 4cosu.du


√(16 - 16sin^2u).4cosu.du

√(16 (1 - sin^2u).4cosu.du

simplify

4√(1 - sin^2u). 4cosu .du remember that √(1 - sin^2u) = cosu

4.cosu.4cosu.du

16cos^2u.du
now integrate this term.

it cannot be integrated directedly as it is cos^2u so use identity cos2x

cos^2u - sin^2u = cos2u

this will give u cos^2u = (1+cos2x)/2



substitute this in place of cos^2u , it will be then

16cos^2u.du

16(1+cos2x)/2.du

8(1+cos2x).du

now integrate this,

8(U + sin2x/2)

the limits are 4 qnd -4 we cannot use them directedly as the term x is also in the form of "u" so convert the limits using this formula.

x = 4sinu

first limit is 4 , means x = 4

4 = 4sinu

u will be π/2
second limit -4 will become 3/2π

now equate using limits.

8(U + sin2x/2)

8[(U + sin2x/2) - (U + sin2x/2) ]

8[( π/2 + sinπ) - ( 3/2π + sin3π)]

8 [π/2 + 0 -3/2π + 0 ]

8π ans
 
leadingguy

leadingguy

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Snowberry said:
Stuck with those 3. :) If anyone could help, it'll be greatly appreciated. Thanks!

http://i.imgur.com/xx9SH.jpg
http://i.imgur.com/Z3VP3.jpg
http://i.imgur.com/Gd0qk.jpg
Click to expand...

first link first part.


sin^2u = x

dx = 2sinu.cosu.du

coming back to question. substitute the value sin^2x in place of x

√(sin^2u/(1-sin^2u))dx

remove square root to simplify

sinu/(√(1-sin^2u)) dx .................. √(1- sin^2u) = cosu

so

sinu / cosu.dx ............... dx = 2sinu.cosu.du (proven above )

sinu/cosu * 2sinu.cosu.du

simplify

/2sin^2u.du this is the fraction proven ANS



2sin^2u.du

sin^2u can not be integratd directly
use identity cos2x = cos^2x - sin^2x

sin^2u = (1-cos2u)/2


now substitute

2sin^2u.du


2(1-cos2u)/2.du

simplify
(1-cos2u).du

now we can integrate these terms directly.

u -sin2u/2

put limits

the first one is 1/4
conert this in the form of u

the formula given is x= sin^2u

so 1/4 = sin^2u

u will be 30 degrees or pi/6

second limit is O convert it using same formula
we will get

O

now using these two limits solve ur obtained expression as follows

u -sin2u/2

pi/6 -sin2.(pi/6)/2 - 0 +sin2(O)/2

pi/6 -√(3/4) Ans
 
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