Questions
The value of is
a
n
b
n!
c
(n+1)!
d
n.(n!)
detailed solution
Correct option is D
Let ∫01 ∏r=1n (x+r)∑k=1n 1x+kdxthe given integrand is differential coefficient of ∏r=1n (x+r)⇒I=∏r=1n (x+r)01=(n+1)!−n!=n⋅n!Alternate MethodConsider (x+1)(x+2)(x+3)⋯(x+n)=etSo that when x = 0, then t = log (n!)and when x = 1, then t = log (n + 1)!log (x+1)+ log (x +2)+ log (x+ 3)+……. + log (x + n) =t∴ 1x+1+1x+2+1x+3+⋯+1x+ndx=dt∴I=∫lnn!ln(n+1)! etdt=etlnn!ln(n+1)! =eln(n+1)!−elnn!=(n+1)!−n!=n⋅n!Talk to our academic expert!
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If then the value of f(1) is
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