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Question:

The value of limx→0 1/x3 ∫0x tln(1+t)/(t4+4) dt is

1/12

1/64

0

1/24

Solution:

limx→0 1/x3 ∫0x tln(1+t)/(t4+4) dt
By applying L'Hospital rule, we get
∴limx→0 1/(3x2) [xln(1+x)/(x4+4)] = limx→0 ln(1+x)/(3x(x4+4))
Applying L'Hospital rule again,
= limx→0 1/((1+x)3(x4+4) + 12x4) = 1/12