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answer..................??????????????????????????????????

Soma Mesh , 6 Years ago
Grade 12
anser 1 Answers
Samyak Jain

Last Activity: 6 Years ago

\int_{-a}^{a}  f(x) dx  =  \int_{0}^{a} { f(x) + f(–x) } dx
So, given integral becomes 
\int_{0}^{\pi/2} {1 / ( [x] + [sinx] + 4)  + 1 / ( [–x] + [–sinx] + 4) } dx  =  \int_{0}^{\pi/2} {1 / ( [x] + [sinx] + 4)} dx  + \int_{0}^{\pi/2} 1 / ( [–x] + [–sinx] + 4) } dx
Let us take first part of the integral.
\int_{0}^{\pi/2} {1 / ( [x] + [sinx] + 4)} dx   =  \int_{0}^{1} {1 / ( [x] + [sinx] + 4)} dx   +   \int_{1}^{\pi/2} {1 / ( [x] + [sinx] + 4)} dx 
                       =  \int_{0}^{1} {1 / ( 0 + 0 + 4)} dx    +    \int_{1}^{\pi/2} {1 / ( 1 + 0 + 4)} dx  
\because for sinx belongs to (0, \pi/2), [sinx] = 0  and  x belongs to (0,1), x = 0 while x belongs to (1,\pi/2), x = 1.
\int_{0}^{\pi/2} {1 / ( [x] + [sinx] + 4)} dx   =  \int_{0}^{1} (1 / 4) dx  +  \int_{1}^{\pi/2} (1/5) dx  =  (1/4) [ x ]_{0}^{1}  + (1/5) [ x ]_{1}^{\pi/2}
                             =  (1/4)(1 – 0) + (1/5)(\pi/2  –  1) = \pi/10 + 1/4 – 1/5  =  \pi/10 + 1/20
                             =  (1/20)(2\pi + 1)           ….....(1)
Now,
\because  [x] + [–x] = –1  \therefore  [–x] = –1 – [x]  and  [–sinx] = –1 – [sinx].
[–x] + [–sinx] + 4  =  –1 – [x]  –1 –  [sinx]  +  4  =  2  – [x]  –  [sinx].
Take second part of the integral.
\int_{0}^{\pi/2} 1 / ( [–x] + [–sinx] + 4) } dx   =   \int_{0}^{\pi/2} 1 / ( 2  – [x]  –  [sinx]) } dx 
                           =    \int_{0}^{1} {1 / ( 2  – [x]  –  [sinx])} dx   +   \int_{1}^{\pi/2} {1 / ( 2  – [x]  –  [sinx])} dx
                           =   \int_{0}^{1} {1 / ( 2  – 0 – 0)} dx   +   \int_{1}^{\pi/2} {1 / ( 2  – 1 – 0)} dx        [Reason is similar as above.]
                      =  (1/2)(1 – 0)  +  (1/1)(\pi/2 – 1)  = 1/2 + \pi/2 – 1 
                      =  \pi/2 – 1/2  =  (1/2)(\pi – 1)               …........(2)
\therefore Given definite integral is : (1/20)(2\pi + 1)  +  (1/2)(\pi – 1)  =  (1/20)(2\pi + 1 + 10\pi – 10)
                   =  (1/20)(12\pi – 9)
                   =  (3/20)(4\pi – 3)   is the answer.

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