∫²₀ (1/x²+4) .solve limits are given

Question

Solve the definite integrals limits are given (2 and 0)

Answer 1

∫_{(from 0 to 2 )} [1/(x²+4)] dx 

 

= ∫_{(from 0 to 2})[1/(4(1+x²/4)]dx

 

= (1/4)∫_{(from 0 to 2)} [1/(1+(x/2)²]dx

 

     Let u = x/2    Then du = (1/2)dx   So, dx = 2du

 

                        When x = 0, u = 0/2 = 0

 

                        When x = 2, u = 2/2 = 1

 

Rewriting the integral in terms of u, we have:

 

     ½∫_{(from u = 0 to u = 1)}[1/(1+u²)]du

 

  = ½Arctanu_{(from 0 to 1)}

 

  = ½(Arctan(1) - Arctan(0)) = ½(π/4 - 0) = π/8

 

NOTE:  You can also use the formula:

 

              ∫[1/(a²+x²)]dx = (1/a)Arctan(x/a) + C, a = positive constant