Question

How do you find the definite integral for: `(1) / (sqrt(1 + x))` for the intervals [0,3]?

Answer 1

`2`

Explanation:

We have:

`int_0^3 1/sqrt(1+x)dx`

Rewrite:

`=int_0^3(1+x)^(-1/2)dx`

We can use substitution here: let `u=1+x`. Fortunately, this means that `du=dx`.

Before performing the substitution, recall that the bounds will also change--plug the current bounds into `1+x`. Thus, the bound of `0` becomes `0+1=1` and `3rarr4`.

`=int_1^4u^(-1/2)du`

Integrate using `intu^ndu=u^(n+1)/(n+1)+C`, where `n!=-1`.

`=[u^(-1/2+1)/(-1/2+1)]_1^4=[u^(1/2)/(1/2)]_1^4=[2sqrtu]_1^4=2sqrt4-2sqrt1=4-2=2`