Question

Value of \(\int_{-\infty}^\infty e^t \,log(1+t)dt\) = ?

(a) Sum of infinite integers

(b) Sum of infinite factorials

(c) Sum of squares of Integers

(d) Sum of square of factorials

I had been asked this question in an online interview.

I'd like to ask this question from Laplace Transform by Properties in division Laplace Transform of Engineering Mathematics

Answer 1

The correct choice is (b) Sum of infinite factorials

The explanation is: \(\int_{-\infty}^\infty e^t (t-t^2/2+t^3/3-….)dt\)

\(\int_{-\infty}^\infty te^t dt=0.5 \int_{-\infty}^\infty te^t dt\)

Now,

\(\int_{-\infty}^\infty te^t dt– 1/2 \int_{-\infty}^\infty t^2 e^t dt + (1/3) \int_{-\infty}^\infty t^3 e^t dt-………\)

Now, \(\int_{-\infty}^\infty t^n e^t dt=n!/(-1)^{n+1}\)

Hence,

\(\int_{-\infty}^\infty t^n e^t dt = 1 – (1/2)(2!/(-1)^3) + (1/3)(3!/)-…….\)

\(\int_{-\infty}^\infty t^n e^t dt\) = 0! + 1! + 2! + 3! +…. = Sum of infinite factorials.