Finding Integral Part of Numbers of the Form a^b Where a Is Irrational
If ∫ xln1+1x ...
Question
If ∫xln(1+1x)dx=f(x)ln(1+1x)+Aln|x+1|+Bx+C, then which of the following is(are) correct
(where C is constant of integration)
A
f(x)=x22
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B
f(x)=x22−12
No worries! We‘ve got your back. Try BYJU‘S free classes today!
C
AB=−1
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
D
AB=−4
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution
The correct option is CAB=−1 ∫x⋅ln(1+1x)dx↓II↓I
Applying integration by parts =x22ln(1+1x)−∫x1+x⋅(−1x2)⋅x22dx=x22ln(1+1x)+12∫x1+xdx=x22ln(1+1x)+12∫x+1−11+xdx=x22ln(1+1x)+12∫[1−11+x]dx=x22ln(1+1x)−12ln|1+x|+x2+C
Hence, f(x)=x22,A=−12 and B=12