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Question

If 3π/4π/4eπ/4 dx(ex+eπ/4)(sinx+cosx)=λπ/2π/2secx dx, then λ is equal to

A
12
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B
12
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C
122
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D
22
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Solution

The correct option is C 122
I=3π/4π/4eπ/4 dx(ex+eπ/4)(sinx+cosx)=λπ/2π/2secx dx

I=3π/4π/4eπ/4 dxex(1+eπ/4x)(sinx+cosx)

I=3π/4π/4eπ/4x dx(1+eπ/4x)(12sinx+12cosx)2

I=123π/4π/4eπ/4x dx(1+eπ/4x)[cos(π4x)]

Let π4x=t
dx=dt

I=12π/2π/2et(1+et)costdt

I=12π/2π/2et(1+et)costdt (1)

Applying king's property,
I=12π/2π/2et(1+et)cos(t)dt

I=12π/2π/21(1+et)costdt (2)

On adding equations (1) and (2),
2I=12π/2π/2sect dt

I=122π/2π/2secx dx=λπ/2π/2secx dx

On comparing, λ=122

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