$\int_0^\infty e^{i \pi x} \left(1-(x+1)^{\frac{1}{x+1}}\right) dx$

[bmmmburns (Marvin Ray Burns A.G.S. (cum laude))]

\int_0^\infty e^{i \pi x} \left(1-(x+1)^{\frac{1}{x+1}}\right) dx

f[x_] = E^(I*Pi*x)*(1 - (x + 1)^(1/(x + 1))); 
g[x_] = x^(1/x); u := t/(1 - t); 

sub = Im[NIntegrate[(f[(-I t)] - f[( I t)])/(Exp[2 Pi t] - 1), {t,
          0, Infinity}, WorkingPrecision -> 100]]

0.1170836031505383167089899122239912286901483986967757585888318959258587743002\
7817712246477316693025869

m = NSum[f[( t)] , {t, 0, Infinity}, WorkingPrecision -> 100, 
  Method -> "AlternatingSigns"]

0.1878596424620671202485179340542732300559030949001387861720046840894772315646\
6021370329665443217278

m - sub

0.0707760393115288035395280218302820013657546962033630275831727881636184572643\
8203658083188126524252

Is the same as 


{Re[NIntegrate[f[t], {t, 0, Infinity}]], and, 
 Re[NIntegrate[f[t], {t, 0, Infinity I}, WorkingPrecision -> 100]]}

NIntegrate::deodiv: DoubleExponentialOscillatory returns a finite integral estimate, but the integral might be divergent.

NIntegrate::deodiv: DoubleExponentialOscillatory returns a finite integral estimate, but the integral might be divergent.

{0.070776, and, \
0.0707760393115288035395280218302820013657546962033630275831727881636184572643\
8203658083188126617723821}

To be continued.

[bmmmburns (Marvin Ray Burns A.G.S. (cum laude))]

Open in Wolfram Cloud ↗

[bmmmburns (Marvin Ray Burns A.G.S. (cum laude))]

I won’t bring the subject of numerical computation into this discussion, but I spent several years learning to compute the digits of \int_0^\infty{e^{i \pi x} \left(1-(x+1)^{\frac{1}{x+1}}\right)}dx.
You can read about my adventure at
How to calculate the digits of the MKB constant .
If you like numeric computations, of much interest is the story of how I came across this integral, by investigating \sum _{x=0}^{\infty } e^{i \pi x} \left(1-(x+1)^{\frac{1}{x+1}}\right) since the 1990s, at Try to beat these MRB constant records!

[bmmmburns (Marvin Ray Burns A.G.S. (cum laude))]

So, I could conclude that

The following "proof of it is from combining

https://www.quora.com/Why-is-it-that-f-x-e-i-pi-x-left-1-x-1-frac-1-x-1-right-int_0-infty-f-t-dt-i-int_0-infty-f-i-t-dt

and

https://www.quora.com/How-would-you-prove-for-g-x-x-1-x-lim-limits_-N-to-infty-int-limits_1-2N-e-i-pi-t-g-t-dt-i-lim-limits_-N-to-infty-int-limits_-0-2N-frac-g-1-it-exp-pi-t-dt.

While

Something similar is shown:

That looks like it proves

to me.
But I don’t see any proof for

.

[bmmmburns (Marvin Ray Burns A.G.S. (cum laude))]

Open in Wolfram Cloud ↗

[bmmmburns (Marvin Ray Burns A.G.S. (cum laude))]

Open in Wolfram Cloud ↗

[bmmmburns (Marvin Ray Burns A.G.S. (cum laude))]

MRB=$\sum _{x=0}^{\infty } e^{i \pi x} \left(1-(x+1)^{\frac{1}{x+1}}\right)
$
vs M2= \int_0^\infty e^{i \pi x} \left(1-(x+1)^{\frac{1}{x+1}}\right) dx in proper integrals

See this notebook.
I got an interesting co-answer here.