Question

How do you integrate `int x/(sqrt(1+2x^2)dx` from [0,2]?

Answer 1

`int_0^2 \ x/(sqrt(1+2x^2}) \ dx = 1`

Explanation:

We want to find the value of the definite integral:

`int_0^2 \ x/(sqrt(1+2x^2}) \ dx`

We can integrate using a substitution:

Let `u=1+2x^2 => (du)/dx = 4x`

So when we substitute we will have `int \ ... x\ dx = int \ 1/4 ... \ du`, and as we have changed the variable of integration we need to change the limits of integration to match:

When `{ (x=0), (x=2) :} => { (u=1), (u=9) :}`

Substituting into the original integral we get;

`int_0^2 \ x/(sqrt(1+2x^2}) \ dx = int_1^9 \ (1/4)/(sqrt(u) \ du`
`" " = 1/4 \ int_1^9 \ u^(-1/2) \ du`
`" " = 1/4 \ [u^(1/2)/(1/2)]_1^9`
`" " = 1/2 \ [u^(1/2)]_1^9`
`" " = 1/2 ( sqrt(9)-sqrt(1) )`
`" " = 1/2 ( 3-1 )`
`" " = 1`