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Choose the correct answerThe Value of `int_(-pi/2)^(pi/2)(x^3+xcosx+tan^5x+1)dx`is
(A) 0 (B) 2 (C) `pi` (D) 1

Text Solution

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Let,`int_(-pi/2)^(pi/2) (x^2+xcosx+tan^5x+1)dx=I ....(1)`
⇒`int_a^b f(x)dx=int_a^b​f(a+b−x)dx`
⇒`a+b−x=π/2−π/2−x=−x`
so,`int_(-pi/2)^(pi/2) ((-x)^2+(-x)cosx+tan^5(-x)+1)dx=I ....(2)`
Adding 1 and 2
=`I+I=int_(-pi/2)^(pi/2) ​0+0+0+2dx`
=`2(π/2+π/2)=2π`
`I=π.`
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