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Question

# If f(x)=limn→∞(2x+4x3+......+2nx2n−1)(0<x<1√2), then the value of ∫f(x)dx is equal toNote: c is the constant of integration.

A
log(11x2)+c
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B
log12x2+x+c
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C
log(112x2)+c
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D
None of these
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Solution

## The correct option is C log(1√1−2x2)+cWhen n→∞, f(x) becomes an infinites series of G.P. with a=2x, r=2x2Hence, f(x)=2x1−2x2Since, 0<x<1√2So, the integral I=∫f(x)dx becomes ∫2x1−2x2dxSubstitute, 1−2x2=t⇒−4xdx=dtAfter substitution, the integral becomes−12∫dtt=−12log(t)+c=−12log(1−2x2)+cI=log(1√1−2x2)+cHence, option C.

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