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Question

If x4+1x6+1dx=tan1f(x)23tan1g(x)+c, where c is an arbitrary constant, then

A
both f(x) and g(x) are odd functions
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B
both f(x) and g(x) are even functions
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C
f(x)=g(x) has no real roots
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D
f(x)g(x)dx=1x+13x3+d, where d is an arbitrary constant.
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Solution

The correct option is D f(x)g(x)dx=1x+13x3+d, where d is an arbitrary constant.
Let I=x4+1x6+1dx
I=(x2+1)22x2(x2+1)(x4x2+1)dx
=(x2+1)(x4x2+1)dx2x2(x6+1)dx
=(1+1x2)(x21+1x2)dx2x2(x3)2+1dx

I=tan1(x1x)23tan1(x3)+c
So, f(x)=x1x and g(x)=x3

f(x)g(x)dx=(x1x)x3dx
=(1x21x4)dx
=1x+13x3+d

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