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

About Integrals Of Some Particular Functions - JKCET

Integrals Of Some Particular Functions is a vital chapter for JKCET aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

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Review the topic breakdown to see which sub-topics within Integrals Of Some Particular Functions carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

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About Integrals Of Some Particular Functions - JKCET

Frequently Asked Questions

Why focus on Integrals Of Some Particular Functions PYQs?

How to best use this analysis?

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

About Integrals Of Some Particular Functions - JKCET

Frequently Asked Questions

Why focus on Integrals Of Some Particular Functions PYQs?

How to best use this analysis?

Chapter Questions
13 MCQs