Question

How do you integrate `{x(sqrt(1+x^2)} dx` from 0 to 1?

Answer 1

`I=1/3(2sqrt2-1)`.

Explanation:

Let `I=int_0^1 {xsqrt(1+x^2)}dx.`

We subst. `1+x^2=t^2 rArr 2xdx=2tdt, i.e., xdx=tdt`

Further, `x=0 rArr t=1, and, x=1 rArr t=sqrt2`.

`:. I=int_1^(sqrt2) t.tdt = int_1^(sqrt2) t^2dt=[t^3/3]_1^sqrt2`.

`=1/3[sqrt2^3-1]`.

`:. I=1/3(2sqrt2-1)`.