(∫_0^1) Cos[x1*x1]*
        E^(-x1^2) \[DifferentialD]x1

N[%]

NIntegrate[Cos[x1*x1]*E^(-x1^2), {x1, 0, 1}]

NIntegrate[
      NIntegrate[Cos[x1*x2*2]*E^(-x1^2 - x2^2), {x1, 0, 1}], {x2, 0, 
        1}]

NIntegrate[
      NIntegrate[
        NIntegrate[
          Cos[x1*x2 + x1*x3 + x2*x3]*E^(-x1^2 - x2^2 - x3^2), {x1, 0, 
            1}], {x2, 0, 1}], {x3, 0, 1}]

Plot3D[
      NIntegrate[
        Cos[x1*x2 + x1*x3 + x2*x3]*E^(-x1^2 - x2^2 - x3^2), {x1, 0, 
          1}], {x2, 0, 1}, {x3, 0, 1}]

Plot[
      NIntegrate[
        NIntegrate[
          Cos[x1*x2 + x1*x3 + x2*x3]*E^(-x1^2 - x2^2 - x3^2), {x1, 0, 
            1}], {x2, 0, 1}], {x3, 0, 1}, 
      AxesLabel -> {"x3", "g(x3)"}]

NIntegrate[
      NIntegrate[
        NIntegrate[
          NIntegrate[
            NIntegrate[
              Cos[x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x1]*
                Exp[-x1^2 - x2^2 - x3^2 - x4^2 - x5^2], {x1, 0, 1}], {x2, 
              0, 1}], {x3, 0, 1}], {x4, 0, 1}], {x5, 0, 1}]

Plot[
      NIntegrate[
        NIntegrate[
          NIntegrate[
            NIntegrate[
              Cos[x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x1]*
                Exp[-x1^2 - x2^2 - x3^2 - x4^2 - x5^2], {x1, 0, 1}], {x2, 
              0, 1}], {x3, 0, 1}], {x4, 0, 1}], {x5, 0, 1}, 
      AxesLabel -> {"x5", "g(x5)"}]

NIntegrate[
      NIntegrate[
        NIntegrate[
          NIntegrate[
            NIntegrate[
              NIntegrate[
                NIntegrate[
                  NIntegrate[
                    NIntegrate[
                      NIntegrate[
                        Cos[x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x6 + x6*x7 + 
                              x7*x8 + x8*x9 + x9*x10 + x10*x1]*
                          Exp[-x1^2 - x2^2 - x3^2 - x4^2 - x5^2 - x6^2 - 
                              x7^2 - x8^2 - x9^2 - x10^2], {x1, 0, 1}], {x2, 
                        0, 1}], {x3, 0, 1}], {x4, 0, 1}], {x5, 0, 1}], {x6, 
                0, 1}], {x7, 0, 1}], {x8, 0, 1}], {x9, 0, 1}], {x10, 0, 
        1}]