Integrals Of Some Particular Functions

%5Cint_%7B0%7D%5E%7B%5Cpi%20%2F2%7D%7B%5Cfrac%7B%7B%7B%5Csin%20%7D%5E%7B100%7D%7D%5C%2Cx%7D%7B%7B%7B%5Csin%20%7D%5E%7B100%7D%7D%5C%2Cx%2B%5C%2C%7B%7B%5Ccos%20%7D%5E%7B100%7D%7Dx%7D%7D%5C%2C%5C%2Cdx

%3D%5Cint_%7B0%7D%5E%7B%5Cpi%20%2F2%7D%7B%5Cfrac%7B%7B%7B%5Csin%20%7D%5E%7B100%7D%7D%7D%7B%7B%7B%5Csin%20%7D%5E%7B100%7D%7Dx%2B%7B%7B%5Csin%20%7D%5E%7B100%7D%7D%5Cleft(%20%5Cfrac%7B%5Cpi%20%7D%7B2%7D-x%20%5Cright)%7D%7D%5C%2C%5C%2Cdx

%3D%5Cfrac%7B%5Cfrac%7B%5Cpi%20%7D%7B2%7D-0%7D%7B2%7D%3D%5Cfrac%7B%5Cpi%20%7D%7B4%7D

%5Cleft%5B%20%5Cbecause%20%5C%2C%5C%2C%5Cint_%7Ba%7D%5E%7Bb%7D%7B%5Cfrac%7Bf(x)%5C%2Cdx%7D%7Bf(x)%2Bf(a%2Bb-x)%7D%3D%5Cfrac%7Bb-a%7D%7B2%7D%7D%20%5Cright%5D

Alternate

I%3D%5Cint_%7B0%7D%5E%7B%5Cpi%20%2F2%7D%7B%5Cfrac%7B%7B%7B%5Csin%20%7D%5E%7B100%7D%7Dx%7D%7B%7B%7B%5Csin%20%7D%5E%7B100%7D%7Dx%2B%7B%7B%5Ccos%20%7D%5E%7B100%7D%7Dx%7D%5C%2C%5C%2C%5C%2Cdx%7D

.. (i)

I%3D%5Cint_%7B0%7D%5E%7B%5Cpi%20%2F2%7D%7B%5Cfrac%7B%7B%7B%5Csin%20%7D%5E%7B100%7D%7D%5C%2C%5Cleft(%20%5Cfrac%7B%5Cpi%20%7D%7B2%7D-x%20%5Cright)%7D%7B%7B%7B%5Csin%20%7D%5E%7B100%7D%7D%5Cleft(%20%5Cfrac%7B%5Cpi%20%7D%7B2%7D-x%20%5Cright)-%7B%7B%5Ccos%20%7D%5E%7B100%7D%7D%5Cleft(%20%5Cfrac%7B%5Cpi%20%7D%7B2%7D-x%20%5Cright)%7D%7D%5C%2C%5C%2C%5C%2Cdx

%5Cleft%5B%20%5Cbegin%7Balign%7D%20%26%20%5Cbecause%20%5C%2C%5C%2Cby%5C%2C%5C%2Cdefine%5C%2C%5C%2C%5Ctext%7Bintegral%20property%7D%5Ctext%7B.%7D%20%5C%5C%20%26%20%5Cint_%7Ba%7D%5E%7B0%7D%7Bf(x)%5C%2Cdx%3D%5Cint_%7B0%7D%5E%7Ba%7D%7Bf(a-x)%5C%2Cdx%7D%7D%20%5C%5C%20%5Cend%7Balign%7D%20%5Cright%5D

I%3D%5Cint_%7B0%7D%5E%7B%5Cpi%20%2F2%7D%7B%5Cfrac%7B%7B%7B%5Ccos%20%7D%5E%7B100%7D%7Dx%7D%7B%7B%7B%5Ccos%20%7D%5E%7B100%7D%7Dx%2B%7B%7B%5Csin%20%7D%5E%7B100%7D%7Dx%7D%7D%5C%2Cdx

.. (ii)
On adding Eqs. (i) and (ii), we get

2I%3D%5Cint_%7B0%7D%5E%7B%5Cpi%20%2F2%7D%7B1%5C%2C%5C%2Cdx%5C%2C%3D%5Cfrac%7B%5Cpi%20%7D%7B2%7D-0%5C%2C%5C%2C%5CRightarrow%20%5C%2CI%3D%5Cfrac%7B%5Cpi%20%7D%7B4%7D%7D

The correct option is(D): 16.5

We have, $f%20%5Cleft(x%5Cright)%3D%5Cleft%5Bx%5Cright%5D%2Bx$
Now, $%5Cint%5Climits_%7B3%2F%202%7D%5E%7B9%2F%202%7Df%5Cleft(x%5Cright)dx$
$%3D%5Cint%5Climits%5E%7B9%2F%202%7D_%7B%203%2F%202%7D%5Cleft%5Bx%5Cright%5Ddx%20%2B%5Cint%5Climits_%7B3%20%2F2%7D%5E%7B9%20%2F2%7D%20x%5C%2C%20dx$
$%3D%5Cint%5Climits_%7B3%2F%202%7D%5E%7B2%7D1dx%2B%5Cint%5Climits_%7B2%7D%5E%7B3%7D2dx%2B%5Cint%5Climits_%7B3%7D%5E%7B4%7D3dx%2B%5Cint%5Climits%5E%7B9%20%2F2%7D_%7B4%7D4dx%2B%5Cint%5Climits_%7B3%2F%202%7D%5E%7B9%202%7D%20x%5C%2Cdx$
$%3D%5Cleft%5Bx%5Cright%5D_%7B3%202%7D%5E%7B2%7D%2B2%5Cleft%5Bx%5Cright%5D_%7B2%7D%5E%7B3%7D%2B3%5Cleft%5Bx%5Cright%5D_%7B3%7D%5E%7B4%7D%2B4%5Cleft%5Bx%5Cright%5D_%7B4%7D%5E%7B9%202%7D%2B%20%5Cleft%5B%5Cfrac%7Bx%5E%7B2%7D%7D%7B2%7D%5Cright%5D_%7B3%2F%202%7D%5E%7B9%2F%202%7D$
$%3D%5Cleft(2-%5Cfrac%7B3%7D%7B2%7D%5Cright)%2B2%5Cleft(3-2%5Cright)%2B3%5Cleft(4-3%5Cright)%2B4%5Cleft(%5Cfrac%7B9%7D%7B2%7D-4%5Cright)$
$%2B%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B%5Cleft(%5Cfrac%7B9%7D%7B2%7D%5Cright)%5E%7B2%7D-%5Cleft(%5Cfrac%7B3%7D%7B2%7D%5Cright)%5E%7B2%7D%5Cright%5D$
$%3D%5Cfrac%7B1%7D%7B2%7D%2B2%2B3%2B%5Cfrac%7B4%7D%7B2%7D%2B%5Cfrac%7B72%7D%7B8%7D%3D%5Cfrac%7B1%7D%7B2%7D%2B16$
$%3D16.5$

High-Yield Trend

Chapter Questions
13 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Integrals Of Some Particular Functions

High-Yield Trend

Chapter Questions 13 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Official Solution

Chapter Questions
13 MCQs