Question

Evaluate the following : `int_(-1)^(1) (x^3 + 2)/sqrt(x^2 + 4)*dx`

Answer 1

Let I = `int_(-1)^(1) (x^3 + 2)/sqrt(x^2 + 4)*dx`

= `int_(-1)^(1) [x^3/sqrt(x^2 + 4) + 2/sqrt(x^2 + 4)]*dx`

= `int_(-1)^(1) x^3/sqrt(x^2 + 4)*dx + 2 int (1)/sqrt(x^2 + 4)*dx`

= I₁ + 2I₂                               ...(1)

Let f(x) = `x^3/sqrt(x^2 + 4)`

∴ f(– x) = `(-x)^3/sqrt((-x)^2 + 4)`

= `x^3/sqrt(x^2 + 4)`

= – f(x)

∴ f is an odd function.

∴ `int_(-1)^(1)*dx` = 0, i.e. 

I₁ = `int_(-1)^(1) = x^3/sqrt(x^2 + 4)*dx` = 0    ...(2)

∵ (– x)² = x²

∴ `(1)/sqrt(x^2 + 4)` is an even function.

∴ `int_(-1)^(1) f(x)*dx = 2int_0^(1) f(x)*dx`

∴ I₂ = `2int_0^(1) 1/sqrt(x^2 + 4)*dx`

= `2[log (x + sqrt(x^2 + 4))]_0^1`

= `2g(1 + sqrt(1 + 4)) -  log(0 + sqrt(0 + 4))]`

= `2[log (sqrt(5) + 1) - log 2]`

= `2 log ((sqrt(5 + 1))/2)`                     ...(3
From (1), (2) and (3, we get

I = `0 + 2[2 log ((sqrt(5 + 1))/2)]`

= `4log ((sqrt(5) + 1)/2)`.