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Question

The value of the definite integral 11dx(1+ex)(1+x2) is

A
π2
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B
π4
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C
π8
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D
π16
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Solution

The correct option is B π4
I=11dx(1+ex)(1+x2)(i)
Using the property: baf(x) dx=baf(a+bx) dx
=11dx1+ex.11+x2
I=11exdx(1+ex)(1+x2)(ii)
Adding (i) and (ii), we get
2I=11(1+ex)dx(1+ex)(1+x2)
=11dx(1+x2)
We know that,
aaf(x) dx=⎪ ⎪ ⎪⎪ ⎪ ⎪2a0f(x) dx, if f(x) is even0, if f(x) is odd
=210dx(1+x2)
I=10dx(1+x2)=tan1(1)=π4

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