Integrate[(x^4 + x^3 - x^2 + 11  x - 12)/(x^2 + 2  x - 
            3), (x)]

Integrate[(x^(-7/3)), {x, 1, ∞}]

Integrate[(1 + 4  x^3) Sqrt[1 + u^2] (1/(1 + 4  x^3)), 
      x]

u = (x + x^4)

D[x*Exp[x], x]

u = 1/(x*Exp[x])*dx

Integrate[u*Sin[Sqrt[u]], u]

Integrate[Sqrt[1 + (y^3 + (1/4) y)^2], {y, 1, 2}]

N[%]

D[Sqrt[ x]/3 - (x^(3/2)), 
      x]

2*pi*
      Integrate[((Sqrt[x]/3) - x^(3/2))*
          Sqrt[(1 + ((1/6) x^(-1/2) - (3/2) 
                        x^(1/2))^2)], {x, 0, (1/3)}]

N[%]

0.037037037037037035` pi* .0015 = Surface Area